I wouldn't even consider it a significant part of my identity. Apply labels to yourself if you choose. Vibe:
A lot of the time "you're not a real gamer" is an accusation thrown at people who don't consider gaming to be their primary identity or worse, people who enjoy video games but are critical of many aspects of the industry, community, etc. 🤔 Should probably worry about that bit.
You do you and the dictionary is gonna do its own thing.