It's probably more "active" sitting. If you are a gamer (especially computer gamer), you are generally not just sitting back "relaxing". Your body is more engaged and you are constantly moving your body in some way.

Sitting and watching tv you can literally be completely still for long periods of time.

Consider StarCraft: Brood War, a legendary real-time strategy game. To be played well, it requires between 200-400 actions per minute (APM), with some players going even beyond 500 APM. Some games last for more than an hour. Players use both the mouse and the keyboard. There's always more to do than you can realistically do. You are always putting out fires, managing your economy, producing units, securing income, carrying multiple attacks at once, fighting tactical battles, and executing strategic goals. Yeah I'd call that a pretty active sitting :)

I am thinking about stance while sitting lately. I am breathing and speaking more from my belly and that starts with posture which is neither slouched forward or back.

Depends on the game. I'd say I have two modes of sitting when programming, one is passive and my muscles ache. Another is active, when I try to use belly muscles (abs?) to keep my posture etc but... When I fall deeply into thinking I will eventually release muscles and feel worse later.

I wonder if there could be an application that would encourage active sitting

They mention reading as an example of active sitting despite the fact that it requires no more motion than changing the channel (or whatever the modern day equivalent is).