Light armour-How about light and medium for the base-class? Subclass can grant heavy for those who need heavy. It seems like a melee-focused class and giving light-armor alone seems like it would make it one of those classes where it only has finesse weapons. Medium as a base can help bridge that gap without the huge buff of heavy.
D10 hit points-Standard for a martial. Good
A spell strike ability (hit with a weapon and have a spell effect go off)-Would this best be expressed through specialty cantrips or does it need to be balanced for any spell stored in their weapon?
Its own unique spell list-Definitely a must to give it its own flavor. Likely a focus on buffs, reactions, and battlefield maneuverability. I'd focus long-range spells on single-target or small bursts and close-range can have more AOE type stuff for damaging spells.
A choice of fighting styles-Also standard for martials. Probably at level 2 for a half-caster.
Half caster with 3 cantrips-As others have said, maybe go with two and get more later. Booming blade/greenflame blade-types seem important. I'd also add here to put in a Fighting Style for another Cantrip similar to the Tasha's Cantrip fighting styles for Ranger and Paladin.
1) Knight - gets heavy armour and defensive spells-Solid. No complaints.
2) A ranged variation (Archer, Dagger thrower etc.) A better-designed Arcane Archer. Uses Medium Armour.-If it's the archer build why is it using medium armor instead of light? Seems like the one most likely to go light armor since maxxing DEX would be the playstyle.
3) An unarmed version based on touch attack spells. Basically an Arcane Monk in light armour. Gets more spells.-Would it be unable to use weapons or would the fighting style just be making it more efficient for it to be unarmed? Also why would this one be the one to get more spells?
Subs discussed elsewhere:
The Psychic: Possible unarmored defense (INT), doing Jedi/Gish type stuff (jump, mage hand, telekinesis, parrying ranged attacks out of the air.)
The Avenger: Divine flavored striker-type. Probably domain spells off the cleric spell list. Named as such from the 4e class and might be better as a paladin subclass but I feel like it fits here better than Paladin.
The Blink Dog: Could fit into any of the other fast and striker-y roles. Goal is maximizing mobility through magic through an at-will teleport (I'd say make it the prime subclass feature and have the distance grow as you gain levels, maybe towards the end allow all movement to be teleportation. Blink spell as a must.
The Channeler: Based off of your #3, the one who focuses more towards magic than weapons. Possibly getting the Wizard's "Arcane Recovery" so it has more spells per day but is not on a different spell-progression than the others.
The Kensei: (change this name obviously) The one who from an outside perspective might look less magical than all the others and uses their spells almost exclusively to buff their fighting rather than anything flashy like shooting fire from their fingertips. Ideally implemented as smaller buffs for the whole battle rather than big spikes like the paladin smite.
The Shadow: The stealth version that stalks their prey before attacking. Somewhere between ranger, rogue, shadow-monk, and Witcher. This might be achieved just within the main spell list and not need to be its own subclass.
I do like the idea of at level 5 instead of getting Multiattack like most martials you get an ability to make a weapon attack as a bonus action if you cast a spell as your action.
For arcane channeling into a weapon I'd say go with an "As a bonus action you can expend a spell slot to begin channeling arcane energy into your weapon. For the next five minutes the weapon is considered magical and deals an additional 1d4 (elemental your choice) damage on a hit (could start with the 5 that are usually the elemental options and then have subclasses grant options like the holy subclass adding radiant and the shadow adding necrotic).
When you use a 2nd level spell slot the extra damage becomes a d6, when you use a 3rd it becomes a d8, 4th d10, 5th d12. This effect ends early if you fall unconscious. You may end this effect early as a bonus action.
At 11th level you can do this at-will as a d4 without expending a 1st level spell slot. At 17th level you get a d6 added on for free.
Might add in an ability that gives advantage on Concentration checks to maintain a spell that targets only you so buffs are more reliable but you don't get to keep a debuff going as easy.
