saying you like something is perfectly fine, but part of being a furry or a gamer or an activist is that in that wonderful little indefinite article you are defining yourself with it, and attributing yourself to it. No, I do not think it is healthy if you have gotten to the point of gaming that you are considered "a gamer" as opposed to a guy that plays games, or a guy that thinks furries are pretty cool.
yes okay that argument seems to be attacking the use of the indefinite article but i swear it's calling out attribution, i promise
Either way, I have no problem with "hobbyist furries" or whatever the heck you want to call yourselves these days to remove the creepy associations. I work on a case-by-case basis. I will generalize but it won't genuinely affect my opinion of a person.
A lot of what you've just said is fine, and certainly people should only claim to be 'a furry' or whatever other title if they are willing to accept the judgements that will bring, even if those judgements are incorrect. However, I think you're still bringing a little too much into the idea of claiming to be part of this group, or any group. I am wary about calling myself a gamer, because I don't play online often, and generally play casually, but if I were talking to someone who didn't play games at all, I may call myself a gamer by comparison; I have clocked upwards of 50 and 60 hours on games I particularly like. Sure, it's spread out over a long period, but that would still seem obsessive to people who never played games.
Point being, there is a lot to explain until people have enough information to gein a "case-by-case" attitude towards anything. I don't really like titles, because as you said people get too serious about it, and let it define them. However, most titles are used to sum up large sections of your personality in a short amount of time. Imagine explaining your musical tastes. You won't list every band you like, you might instead say "I'm a bit of a punk/mosher" or "I mainly listen to pop/classical/rap" or whatever. The same logic can be applied here. People who consider themselves 'furries' obviously consider whatever aspect of furry they identify with to be a large part of their personality, and so they are attempting to explain their personality in a small amount of words. It will never be fully accurate, but even your nationality is not fully accurate. I would describe myself as English, when I'm actually also part Irish. If I decide to accompany that and call myself British, I need to take into account my small amount of Spanish heritage, and so on and so forth. Nothing is clean cut and black and white, but taking everything into account would be impossible.
However, even with all that, yes. There are people who a) let it define them when really they probably shouldn't and b) People who call themselves furries and then get angry when people make assumptions that, in their eyes, are perfectly reasonable.