The problem stems from people wanting to (for some reason) specifically define what a "gamer" is.
I personally am of the opinion if you play games on whatever device or platform, you're a "gamer".
Though if you play board games or card games and call yourself a "gamer", I don't see anything wrong with it if you feel like labelling yourself as such. Just take note that general usage of "gamer" would typically refer to games played over digital mediums. In a conversation where you tell someone else you're a "gamer", they would most likely assume you mean you play computer/console games, so it would probably be best to clarify that you play card/table-top games etc.
Whether or not the conversation will devolve into a debate about what constitutes a "gamer" however, I can't say!