we need to find ways to say this that are not insulting.
They are wrong, it is beyond a doubt provable false, becuse there is nothing in common with spells at all. The closest is if you say EVERY special ability (flurry of blows, rage, power attack, dodge, aim, sneak attack) is a spell... and at that point you are twisting the entire system to a breaking point.
I would love to see you try to prove that this is provably false.
First of all, enough of the straw man arguments. No one is saying power attack, aim, or sneak attack is a spell.
Martial mechanics get called spells if they share a sufficient number of qualities with spells. Those qualities would be things like:
a) Time boxed into a certain number of uses per time period. It's particularly egregious when for example you have two equally strenuous combat maneuvers each of which you can only use 1/day or 1/encounter or something. It is easy to explain at the metagame level why this constraint would exist but not at the in game level without a lot of jumping through hoops. Spells often have limitations like this that are "just because" but we accept them (usually) because "magic".
b) Disassociated from a mundane fictional explanation. That is to say, if the mechanic lets you do something that isn't a plausible result of exceptional strength, skill, and speed, and in particular if you can do something according to the mechanic which people of similar strength, skill and speed can't because of an esoteric technique then you have a spell. Examples would be conjuring ammunition into existence at will, stomping on the ground to create an earthquake despite having mundane strength, conjuring elements like fire and lightning into existence with your attacks, ringing your sword against stone to create a powerful sonic attack, etc.
c) Causing effects to happen without contests between the attacker and the target. That is, something happens in the game fiction which could be plausible for the "mere" application of exceptional strength, speed and skill but for whatever reason it always happens regardless of the strength, speed, and skill of the opponent.
d) Explicit flavor texts or mechanics that resemble the flavor text associated with spell-casting classes, such as needing to prepare stances and maneuvers at the beginning of the day from a pool of known maneuvers, having levels of maneuvers from 1st to 9th, having prepared maneuver slots, etc.
So for example, suppose a martial character has to spend 20 minutes practicing his maneuvers to prepare them at the beginning of the day, and he can only perform maneuvers he prepared even if he knows other one, and he has a single 6th level maneuver slot that the decides to fill with a maneuver called "Sonic Boom" that lets him use his weapon to produce a line of sonic energy that does 6d10 damage to all in 60 feet and pushes them back 5 feet, then I think it's fair to call that mechanic a spell even if it is a martial character that is doing it. That's obviously a somewhat extreme example (but there are real examples of the same type in D&D's history), but if the mechanic looks something like that, then don't be surprised if people say, "Effectively, adding that ability to martials is just giving them spells."
Of the mechanics you list, then only one that is spell-like is "Rage", and I'm OK with people seeing Rage as a sort of spell because if you look at the real world origins of the archetype then you are often looking at people who practiced some sort of ritual magic to induce a battle frenzy in themselves and who believed that there was some sort of magic involved (for example it was common to believe across several cultures that the supernatural forces of that culture would harden the warriors skin to make them damage resistant if proper rituals were performed). I won't list a lot of examples out of a desire not to derail the thread with questions of cultural sensitivity, but the archetypal example was the berserkers of Norse culture that believed they were taking on the spiritual and physical power of a bear (and may have been getting themselves high on hallucinogens in order to deaden pain and the shock response from being injured).
