Command Presence
The mere presence of a Warlord within an adventuring group instills confidence and facilitates synergistic cooperation. The Warlord’s competent direction imparts a sense to all their allies that someone always has their back. This gives allies a boldness to take chances they normally wouldn’t, and do so with sureness they normally wouldn’t possess.
Once per round, any one member of the Warlord’s group can apply advantage to any one roll – be it an attack, a save, or a check – but the group must collectively agree to its use. This is not cumulative and cannot be carried from round to round; each round has one and only one usage for the group.
This catches the idea of Commanding Presence, and sounds good in concept, the actual mechanic, though broadly applicable and not lacking in effectiveness (gaining/granting advantage usually takes an action), is pretty bland compared to Commanding Presence, which gave each flavor of warlord it's own thing.
Inspiring Word
A Warlord builds a significant rapport with each of their comrades, learning what motivates each one of them. They know their compatriots so well they can rejuvenate and refocus them with just a word.
Once per short rest you can encourage or call out to a faltering companion and bolster their resolve; even allowing them to shake off some of the accumulated wear and tear of combat. Doing so either allows an ally to recover hit points equivalent to the Warlord’s Wisdom or Charisma modifier times ½ the allies level (minimum of 1), or free an ally of non-magical fear (the Frightened condition), or reduce any effects due to exhaustion by one level.
The ally must be within hearing range of the Warlord, and cannot be at 0 hit points. At 5th level they can do this two times per short rest, three times at 10th level, and four times at 15th level.
You might just as well not have hp restoration if you can't use it to stand up a fallen ally. Though some h4ters made a point of asserting that 0hp = unconscious and unconscious = deaf, even though the latter is demonstrably false, even if that point had been valid, it'd be worth it to bend credulity a bit in the name of fantasy heroics and playability and let it work on dropped allies.
Also, this is one of those places where 4e and 5e did things pretty differently. In both 4e & 5e, hit points are a daily resource that face attrition over the course of the adventuring day. In 4e, natural healing in the form of surges was the major source of hps, and Inspiring Word (and most literal 'healing' as well) was an encounter-resource surge-trigger, rather than a hp resource in itself. In 5e, HD are a less significant hp resource, with daily hps, consumable potions, and spells being far more important in managing hps. This version of Inspiring Word is 'too 4e' in being a short-rest resource.
I do like it being able to affect exhaustion, though.
Call of Restoration
With the same rapport they use to inspire their allies, they can even sometimes call them back from the edge of death itself.
Through a combination of vocal exhortation and physical shaking, you can attempt to rouse a fallen ally (at 0 hit points). They must be able to hear you (not Deafened) and must be adjacent to you. The ally can immediately make a death saving throw, even if they have already done so for the round, and make the roll with a bonus equal to the Warlord’s Wisdom or Charisma modifier. If successful, the ally is returned to consciousness and recovers hit points equal to the Warlord’s Charisma or Wisdom modifier times ½ the allies level (minimum of 1).
You may attempt this once per long rest. At 7th level you may attempt this twice per long rest, and three times at 17th level.
So the Warlord can stand up an adjacent fallen ally once at 1st level, all the way up to thee times at 17th. A cleric with Cure Wounds prepped, OTOH, can do it three times at first level and well, a whole lot at 17th. That's some very strict strict-inferiority there.
I get what you're trying to do here, though, I think: you're trying for one specific bit of genre-emulation in the distinction. I don't think it's quite necessary. Let Inspiring Word restore from 0, remove the explicit hearing/shaking requirements, maybe link it to a 'command radius' or something. Then, note in a side-bar that DMs may decide what sort of 'inspiration' seems appropriate to them for the situation and the tone of the campaign their running. It might run from a character being 'inspired' even when the warlord can't act or isn't present (he just remembers something inspiring) to inspired by example, to vocal exhortation, to physical shaking.
Also, I'm sensing StatMod*level/2 as a standard, here. I'm not sure it's a great formula. Proportional to level is good, certainly. OTOH, HD (whether expended or not) could be a more proportional-to-class standard (it seems like it'd be fair for an inspired barbarian or fighter to get more hps than an inspired rogue or wizard). Stat mod as a multiplier rather than an adder, I'm not so sure about, either - has 5e done that anywhere else?
Tactical Leader
A leader can unify and focus a group, making a group greater than its parts; but a leader that doesn’t have the trust or cooperation of their compatriots is less than useful, and becomes a burden to the group - a catalyst for dysfunction.
A leader may give up any or all of their actions to be used by any ally; especially move and attack actions, and including bonus actions, and reactions (if appropriate). These actions can be used by the ally even if they have already acted in the round, and may still use their own unused actions after using the Warlord’s actions. Any actions given in this manner must be used in a manner consistent with their original purpose (move action must be used as a move action, etc.).
The Warlord player must designate what the action or actions are to be used for (such as “Aleric is in trouble - disengage from your foe, and go back him up”), and the character receiving the actions must use them consistent with the direction (the DM can determine if consistency is maintained if a receiving character’s player “calls an audible” or otherwise alters or interprets the “spirit” of the directions).
Allied players can also ask the Warlord player for their actions; such as, “Do you want me to back-up Aleric?” But the Warlord player still designates what actions they are giving up and what they are to be used for.
Players can disregard the directions of the Warlord’s player (“Aleric can take care of himself!”), but then those actions the Warlord gave up are lost for the round (they still expended the time issuing directives).
If players consistently disregard the Warlord's directions (DM discretion), then the DM may decide that the group has also lost synergy and can no longer benefit from the Warlord's Command Presence.
Almost strikes me as a better guide to designing specific maneuvers (or whatever) than a general ability. It's certainly flexible enough, in as far as 5e goes in providing possible actions, which is potentially pretty cool, just like the Cunning-Action-like ability, below.
Again, I think you're going a little too deep into the tone of the fiction side, but at least you're keeping it 'DM discretion.'
Oh, and a potential danger (or feature) of it being this general is that the Warlord might easily end up with his own actions not ever being worth taking. That was something you could do with the warlord, intentionally, with specific build choices, but this could have it happening to any Warlord as a matter of course...
And, while the action economy management is cool, action economy is less important in 5e than it was in 4e - and management of daily resources much more important. So, for instance, it's potentially effective for the Warlord to have a caster use his action to cast an extra spell, but that spell is still coming out of the casters' limited slots - and if the Warlord were just replaced with another caster, you could do the same, and have twice as many slots.
Bonus Action
Beginning at 6th level, a Warlord’s quick thinking enables them to move, act, or give direction more quickly. You can take a bonus action each of your turns in combat. This action can only be used to take an Attack, Command Action, Dash, Disengage, Dodge, or Help action.
At 13th level this increases to two bonus actions per turn.
Like the Rogue's Cunning Action, this is kinda cool, a simple mechanic that opens up a fair range of tricks. Obviously you can't call it 'Bonus Action,' as that's already the action type. It should be phrased 'use your bonus action to...' since everyone does get a bonus action every turn, just doesn't always have something they can do with it.
Action Inspiration
You can inspire others through calm reinforcement or inspirational counsel. Once per short rest, you can inspire an ally attempting an action to greater success. That ally can add the roll of a d6 to their check roll. The action that the inspiration applies to must be specifically stated (picking a lock to get the group through a door, crafting an item for a mission, disabling a trap, etc.).
That's very like the Inspiration mechanic, which is fine for verisimilitude but maybe weakens said mechanic (which is one of those carrot-for-RP tricks) - assuming it doesn't stack, which'd make sense. A random bonus gives an obvious comparison to Guidance, which is unlimited-use, as a Cantrip, but only d4. Nothing would stop the two from stacking, either.
Strategic Leader (in-work)
The Warlord has learned how to apply their leadership skills and expanded their combat knowledge to include leading large groups. At 11th level and once per long rest, they can apply advantage to an action that a unit larger than their group, but no bigger than a company, undertakes. At 16th level, they can do this twice per long rest with a unit larger than their group but no bigger than a company, or once per long rest with a unit larger than a company up to the size of a battalion/regiment. At 20th level they can do this three times per long rest for a unit larger than their group but no bigger than a company, twice per long rest with a unit up to regiment/battalion size, or once per long rest with an entire army.
(explore faster than normal march, advantage to checks for fatigue due to a forced march, etc.)
…look at mass combat rules…
Not sure this'd come up. Large numbers tend to break 5e due to bounded accuracy, anyway.
Rally the Troops
Once per long rest, a Warlord can motivate and focus their group with a stirring speech (whatever group they are leading at the time, be it their adventuring group, a company, or a whole army). Doing so either allows all members of the group to recover hit points equivalent to the Warlord’s Wisdom or Charisma modifier times ½ the allies level (minimum of 1), or free all members of the group from non-magical fear (the Frightened condition), or reduce any effects due to exhaustion by one level.
The group must be within hearing range of the Warlord, and only affects members not at 0 hit points.
Any hp-restoration or hp-management that can't stand up an ally at 0 hps is decidedly inferior. I can't tell if this is supposed to be an in-combat Action or an out-of-combat function like Inspiring Leader, and that makes a big difference - 'rally' really implies in-combat.
Command Actions (in-work)
Individual tactics or maneuvers - based on Battlemaster maneuvers, 4E Warlord powers, etc.
This'd have to be the meat of it, where player choice and character versatility can be provided. The Warlord had a lot of exploits, not quite as many as the fighter, but hundreds, and they did a wider variety of things. In addition, leadership could impact more than just combat. It'd be easy to throw a bland, non-stacking, mechanic like advantage, and doing so wouldn't lack in effectiveness, but it'd be pretty dull, like a Cleric who does nothing but cast Bless (which, according to some game reports we've heard around here, isn't much of an exaggeration).
5e has put a lot of it's eggs in the spellcasting basket, a huge chunk of the book is devoted to spells and 30 of 38 builds cast spells, leveraging that resource to give them versatility, choice & effectiveness without necessitating a lot of page count in each class, individually. Since the Warlord needs comparable ability, but can't leverage the game's deep investment in spells, it'd need a surprisingly 'long' write-up...