I've never seen "gamer" as being someone who just plays games, but someone who defines some facet of their personality around gaming. Where it starts to bleed into other aspects of your life - the sort of people who have video game-related in-jokes with friends, or lots of video game merch, or basically sort of "living" games rather than just playing them. "Gamer" is when gaming stops being an activity and becomes part of your life (and it's not necessarily a competitive mindset - I don't think you need to be a "pro gamer" sort of player to be a "gamer").
For me personally the two have always sort of been inextricably intertwined - gaming came into pop culture right around the same time I started gaming, so it was easy to fall into that sort of mindset where gaming was a huge part of my identity. When I was around 5, my cousin had a ColecoVision and I used to like to play it when I was at his house, then I got one for Christmas. I played Coleco and Atari 2600 games for years, then moved on to the NES and SNES, then the PS1 and PS2, the Xbox and the 360, and now the PS4. There's never been a time in the last 40 years when I didn't have at least one console hooked up in my house (or college dorm). Sometimes multiple ones at once.
Meanwhile, I was pretty much raised on the massive video game themed assault on pop culture in the early 80s. Shows like Saturday Supercade, Pac-Man, Pole Position, and the later Super Mario Bros cartoons. Pac-Man shaped Spaghetti-Os. Donkey Kong and "Nintendo Entertainment System" cereal. Board games. Movies. Even non-game related stuff like Tron that was game-adjacent. I dressed up like Link for Halloween. I've owned Nintendo-themed t-shirts, Atari-themed t-shirts, Final Fantasy themed t-shirts, and even a Halo t-shirt. I've had video game themed keychains. I have video game themed stuffed animals.
I'd play arcade games like Arkanoid, Rampage, and Gauntlet (and later Street Fighter) when my mom took me to the grocery store to shop. Later I'd go to a local bowling alley every couple of weeks and play games with my friends (off the top of my head, I remember Dragon's Lair 2, Time Traveler, Hard Drivin', The Avengers, X-Men, and eventually Mortal Kombat). A few years after that, I'd occasionally go to arcades at the local boardwalk and play games like Revolution X or Mad Dog McCree. As an adult I've taken my niece and nephew to Chuck E. Cheese and played games like Mario Kart or Let's Go Jungle! with them.
If you asked me to imagine a world where video games were never invented, I'm not sure I could - so much of my life has revolved around gaming in some way I can't imagine what it would be like without it. That's sort of what it means to be a "gamer" to me.
And it's basically been a core part of my entire life. Along with comic books and fantasy novels/shows, it defines a large part of my interests and hobbies. And has done for almost as long as I've been alive.
This is stupid AF to me, a true gamer to me just means someone who has played a variety of games throughout their life and actively plays a good amount (has it as one of their main hobbies). You don't have to be materialist and buy tons of BS merch. The games or consoles themselves were good enough, and you might not even own them anymore but you did at some point. You don't have to make it your entire identity or personality if you don't want to, but you should be able to talk about, discuss at length, and critique games to a good extent.
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Playing: TotK, XB3, TotA, Afterimage MHR:SB, MHW: IB | MR329+/HR 561+ | LS/CB/GS/BOW "The imagination is a weapon, those who don't use it, die first"