I feel that the distinction you are suggesting is a bit too specific for most practical purposes.
Essentially, everything you mention falls under the heading of martial arts, in my book.
Of course, if you are training in one of the more traditional arts that focus on psycho-physical health and self-improvement, such as Aikido or Taiji, you should know that they won't prepare you for a self-defence situation without additional training. (Not that there would be anything innately wrong with the way they are normally taught - not everyone's goals are the same, after all.)
If your training is self-defence based, it should include skills like situational awareness, stress management, de-escalating a situation, and so on, to begin with. Just teaching the moves that would (hopefully) handle a specific attack isn't going to cut it!
For that matter, the skill set to be taught to law enforcement officers or bouncers (who are expected to be able to control an aggressor without inflicting lasting damage) must be different from the one in a women's self-defence class (where the "evade and destroy" approach is generally the most viable option).
Obviously, if you are training to win tournaments, that's a different animal altogether. Despite all the talk about MMA being the ultimate test of fighting prowess, fact is that what might go down in a self-defence situation is very different from what is happening in the octagon.
So yes, be clear what your training is all about (and it could fall into more than one category, to be sure), but don't overspecialize. Try to be inclusive rather than exclusive! Aim to become a well-rounded individual in what you do.