You are making tons of false equivalencies.
I gave this example before but I'll give it again, my mom plays some world game on her phone. She wouldn't call herself a gamer. Because gamer has a certain connotation and it isn't "everyone who plays games". As much as you want it to mean that it simply doesn't.
The gamer term is associated with "gamer culture" without a doubt. There's a reason it's called gamer culture to begin with.
And, as we have established, I read books but I'm not a bookworm or a book reader because reading is just a normal thing that, at a certain level, doesn't define you. I also listen to a ton of music but I have no name for it and if I do is associated with the type of music I listen to and not the fact that is music.
Making comparison to sports (for some reason) or with stuff that has nothing to do with that doesn't make what I'm saying less true.
The term gamer has a meaning and a weight behind it even if you don't want it to have. My dad played all Tomb Raider games, if you ask him he wouldn't say he is a gamer. Because that term isn't used in that context. It's exclusive and it makes it seem like you have to define yourself with video-games to even enjoy it. Which is bollocks and not something that anyone should defend.
If you're going to say my examples don't count the very least you could do is explain why.
It doesn't matter what your dad calls himself, that has 0 effect on what the word means. It literally means 'someone who plays games'. I don't get how you think there's anything more to it than that. Someone who gambles is a gambler. It doesn't matter if they only gambling they do is once a month playing poker with friends, all that matters is that they gamble. If someone had brown hair, they're a brunette wether they call themselves that or not.
It's exclusive and it makes it seem like you have to define yourself with video-games to even enjoy it. Which is bollocks and not something that anyone should defend.
This is the part that continues to blow my mind. It's got nothing to do with defining yourself. People keep saying that and it's utter nonsense. If I describe myself as tall, I'm not saying or implying that being tall is what makes me who I am. If so said I was a stamp collector, that doesn't mean I don't have other hobbies or interests. Or if I say I'm blonde that doesn't mean I'm dumb because of a stupid stereotype. You are literally changing what the word means just to suit the argument. It is a description, not a bloody label that defines who you are as a person like you're making out. It's a term that means one specific thing, just like many other terms like golfer, thrill seeker, metal head, and thousands of others. Just because we don't have similar words for someone who watches movies or reads books doesn't mean we should stop using this one. That's like saying we should stop calling football players footballers because we don't call tennis players tennisers.
You are the one that has this weird, mind boggling view that some how this one word has this deeper, and potentially negative meaning when t is absolutely no different than the thousands of other words we use daily to name the collective group of people who tick a certain box. Pensioner, brunette, theif, driver, foodie, socialite, carpenter, painter, guitarist, photographer, gamer. They all just mean 'someone who ticks this box'. You thinking that gamer is different and somehow means more, or something especially negative, is your own problem. It's nothing to do with the actually word or what it means.