another specifically calls them out as a type of "non-traditional"-magic
This is not true. It says that they are not magic in the traditional sense. That neither asserts nor entails that they are some other, non-traditional sort of magic. It leaves the matter open. That is a deliberate point of drafting.
If they wanted to write "Martial powers are magical, but not in the traditional sense" they could have. But they didn't.
That appears to be exactly what this text is saying, since it is directly comparing those things to arcane formulas and prayers, which are.
Yes. It denies that martial power involves those things. Instead it draws upon willpower, strength and training.
Seems like a terminological issue to me; it might be supernatural (in the literal sense of being above and beyond what is natural) but not magical (in the D&D-specific sense of referring to discrete powers granted in some organized form by specific entities or forces).
EX abilities in 3E are defined as "not magical though not physically possible either". What does that mean in the fiction? How are these physically impossible things achieved? The game doesn't tell us. The only practical significance of the denial that a troll's regeneration is magical is that it keeps working in an anti-magic field.
4e Martial abilities would keep working in an anti-magic field too.
Total, obvious (not zap/boom magic), unless you are feigning English reading comprehension issues, which, to be honest, I thinks is the case for some, ya know, gotta keep the agenda/crusade burring.
You do realize that it goes on to define what it means. right?
herrozerro is obviously right. The assertion "Martial powes are not traditional magic" naturally gives rise to a question "What are they, then?" The paragraph goes on to explain: they are extraordinary abilities, going beyond ordinary mortal capabilities, driven by willpower and training and strength. Whether a given play group wants to treat that as subtle magic (as some apparently see Aragorn) or as Conan-esque capabilities is left open. It could be different for different PCs in the same game. It could be different for a single PC over his/her career - a player might begin conceiving of his/her PC as "mundane", but as s/he gains levels and becomes a demigod, the player might change his/her conception of his/her PC's source of power.
I don't think anyone would argue that even though Hercules didn't cast spells his strength wasn't magical
And to that I'd say, that high level martials have always been magical.
I have two versions of the game that flag Hercules and comparable legendary figures as paradigm fighters: Moldvay Basic ("Great heroes such as Hercules were fighters": p B10), and 2nd ed AD&D ("There are many famous fighters from legend: Hercules, Perseus, Hiawatha, Bewoulf, Siegfried, Cuchulain": p 26 of the fighter-on-horse printing PHB).
It's always been true (at least through B/X and AD&D; I don't know about 3E/PF) that a high-level D&D fighter might go toe-to-toe with Demogorgon or a great dragon, and otherwise do the sorts of things that Hercules did. If that means "magic" for some players, that's hardly something that 4e introduced to the game.
You can start right in the PHBI with the fighter power that allows them to do triple (or 3[W], or whatever) damage once a day. In every relevant sense, this is a magical ability. In a previous edition, it would have been referred to as a smite attack and been given appropriate mechanical descriptors. In 4e, it's a martial power. In no edition is that not a magical action.
This has been discussed at length upthread. 4e uses different rationing techniques from 3E. In particular, it uses player decision as a rationing technique. That does not mean that the ability is magical. A fighter doing 3W damage from a daily power is no more magical than a fighter doing double damage from a critical hit. It's just that the game shifts the rationing mechanism, from lottery to player choice from a pool of resources.
If you don't want to play a game where non-magical abilities are rationed via a method other than lottery, that's of course your prerogative. That doesn't mean that the rationing mechanic changes the fiction of the game. There are plenty of other RPGs out there which use player-driven rationing but don't have magical PCs. (For instance, in MHRP the Punisher's abilities are rationed by player choices about resource deployment, much like a fighter daily in 4e, but this doesn't mean that The Punisher in that game is a magician rather than a skilled, and lucky, soldier.)
