ya as much as anyone wants to put it as far as the sca, living history, fantasy etc, we're all LARPS. i dont care what people say it's true. we all play a game and dress up. whether it's to honor vets, get rid of reality or what have you. just different names and different faces.
I guess this is where the big difference lies between all the groups you listed and HEMA/WMA. It's first and foremost a martial art. It encourages no role play, no personas, and many schools don't require any period clothing unless used for purely functional purposes. That's why they can get upset when someone comes along and suggests to them how to improve upon their martial art when that person is coming from a role-play perspective.
The line is skewed only because we view WMA/HEMA in the same light as the other mentioned groups because they use swords and other medieval weapons. Do you look at Kung Fu or Karate schools as LARPs just because they wear traditional garb? No... because they are martial arts schools, teaching a combat sport. So it is with HEMA/WMA school. They really shouldn't be lumped in with Living History, SCA, Boffer LARPs etc, because the aim is to teach a combat sport, not role play or be a persona etc... It gets blurry here because the SCA has a sporting aspect within an overall role-playing community.
But you have to be respectful of the WMA crowd's perspective here. the analogy would be if a Kendo school was approached by a group of people who re-enact Samurai era history and suggested technique to the Kendo guys. I think we can see here, that it simply doesn't make sense, and the Kendo school might even be dismissive of what the re-enactors are trying go say. So it is with the WMA community and the SCA. Let people have their fun, but don't be surprised if one group doesn't want to listen to the other. The point is WMA, although historical in context, is a martial art, not a LARP, and I think that's the point that people are overlooking when it comes to this particular topic.