I think my mom playing Wordle on her smartphone does indeed make her a gamer.
It's a lot more useful though when words have meaning and we take them seriously, otherwise nothing really makes sense.
Humans by their very nature use games to pass time and stimulate their mind and body, it's one of the most universal things about us.
That game of Wordle my mom plays once per day is probably what keeps her paying a subscription to NYT. That's real money. She'd probably drop the subscription otherwise.
NYT sells a games-only subscription for $50/year, $80/year per family. I think NYT executives would be happy to consider someone who plays games for 5 minutes per day or even 5 minutes per week a gamer.
The $50 I spent on Baldur’s Gate 3 is the same $50 as a year’s subscription to Wordle (and the other NYT games).
“Gamer” may refer to a stereotype but there is fundamentally no difference in concept.