Hey, Tony,
I don't really think it's that circular, honestly. If we were only looking at single systems in isolation, then yeah, I'd be with you… however we have many, many, many "Traditional Martial Arts" that we can compare and cross-reference. But my point is that accepting that someone says what they do is "TMA", when it plainly isn't, doesn't do a thing when it comes to defining what TMA actually are.
Take a look at the fake "ninjitsu" systems out there… they're not in any way genuine, they're not in any way "traditional", they're not in any way Japanese in most cases… should we class them as actually "ninjutsu" as part of an effort to define the art, just because some deluded people think that's what they're doing?
How about if someone trained in a bit of Judo, did some Arnis, added some Capoeira, and called it BJJ? Do we accept that that (hypothetical) "art" actually is BJJ? Or do we compare and contrast it with known BJJ to properly define and categorise it?
The problem is that there is no universal (or even mostly universal) agreement about which arts fall into that box of many, many, many traditional arts. If we mostly agreed upon which arts fall into the TMA category, then we could derive a definition from observing the common characteristics of the examples in that category. If we mostly agreed upon a definition, then we could decide which arts to include or exclude from the category based on the definition. Since we don't have common ground on either starting point, then it's silly for any one of us to declare that our definition or our categorization is objectively the factually correct one.
With regards to your hypotheticals, there is universal acceptance that "BJJ" is the term for the (mostly grappling with a heavy emphasis on groundwork) art coming out of Brazil in the 20th century developed by the Gracie family (and their students) (and Luiz Franca and his students) from an initial introduction to Judo as taught by Mitsuyo Maeda. If someone was to call an unrelated art "BJJ" they would be using the term in a way unconnected to any common usage of the term and would probably be doing it in a conscious attempt to fool people.
"Ninjutsu" gets a little more complicated, since the common usage covers a few options.
There are the actual historical Japanese espionage arts, elements of which may or may not survive to the present day.
There are the X-Kans and their derived splinter organizations, some of which use or have used the "ninjutsu" label even though only 3 of the 9 Takamatsuden arts are identified as ninjutsu and none of those 3 are known for sure to go back further than Takamatsu himself.
There is the movie image of the black-clad assassin disappearing in a puff of smoke.
In the case of folks claiming to train ninjutsu, I'd look at what they're actually claiming their art to be.
If they are claiming to teach a specific historical Japanese art, then I'd look at whether they can back that claim up. (Probably not.)
If they are coming from the X-Kans or their spinoffs, then I'd kind of prefer they didn't call it ninjutsu, since even if the 3 arts in question are legitimate, they make up only a small percentage of the X-Kan training. Still, I'll give them a pass since they might possibly have a link to an actual historical ninjutsu tradition or three.
If they claim to be teaching the fantasy as shown in the movies, then they are delusional or liars since that doesn't exist in the real world.
If they claim to be teaching something else and just using the term ninjutsu, then I would point out that they are using the word incorrectly since it doesn't match any commonly accepted usage.
"TMA" is much more problematic because we don't have agreement on what the term means or which arts fall into that bucket. I commonly see TKD referred to as a TMA and BJJ referred to as not a TMA. How does that make any sense when BJJ is older than TKD? BJJ has evolved, but not any more so than TKD. You can find instructors teaching old-school BJJ just as easily as you can find instructors teaching old-school TKD.
Probably most people would accept the koryu arts as being TMA, but once you move past them you're going to find very little universal agreement.
Depending on who you ask, a TMA might be:
An art that has existed for a certain length of time.
An art that has existed unchanged for a certain length of time.
An art that has existed unchanged since its formulation by its founder.
An art with certain cultural signifiers (rituals, attitudes toward instructors, etc)
An art with an accurate historical record and unbroken lineage of instructors.
An art with a certain set of values (which may vary according to who is offering the definition)
An art with a certain set of training methods (which may vary according to who is offering the definition)
Personally I don't care which definition you use, as long as you make it clear beforehand so I know what you're talking about.