Well, let's use that same logic to argue that almost anyone can (and does) teach their impression of Aikido as "explaining an idea as they truly understand it". They "get people to do Aikido" and wear black culottes... who can tell the difference? And so on. Right now, I'll guarantee you that there are a number of different takes being taught by a number of different people as "aiki". Are they all teaching the students what could convincingly be taught as passable jin/kokyu skills in classical martial-arts? They can't be. So yes, there's a reason to try to mesh terms with the accepted definitions.
Remember the old "Teacher Test" I did that everyone made too much noise about? What happened was that after listening to some self-styled "Xingyi" teacher drone on an on about himself for 4 hours, in desperation I asked him to place his palm on my (right) chest and hit me without pulling his hand back. So he did. And it was very obvious that after all the talk about "internal", his ability to hit was still mostly shoulder derived. Not dantien/hara. That immediately tells us about all the rest of the stuff he does and logically, he doesn't really do justice to a so-called "internal" martial art. Would you suggest that his take on "internal" was a valid one?
We're agreeing here in this scenario, just to be clear about that and get it out of the way.
Yeah, at some point you can't just telepath all communication between two people, and you do need to be able to share some ideas. But if the concept the words are pointing at has not actually been shared between the conversants, words can only do harm (song quote very intended). So where and when is terminology useful? And yah, as you point out, one test is worth an unlimited supply of words...
If he just showed what he was doing and didn't try to label it as anything, the misunderstanding would have never happened. He moved one way, you would have liked him to move another way to agree with a certain definition. I think I am just saying that "internal" might be too broad and reductive a term that it will inherently always create such misunderstandings. Terminology might be better left for scenarios that are easier to verify. Instead, maybe we are better off training together and sharing that way so we clear up misunderstandings sooner before they are roaring debates and create animosity.