I'm pretty sure for skaters it's games played in the regular season. Sitting in the press box doesn't count. For goalies it's obviously a different story due the nature of the position.
My initial thinking before I looked up the rules was that they could petition the NHL to have Cote on but probably would not for Backlund... After learning the rules I see that they couldn't petition them for Cote but could for Razor due to the circumstances. And that Backlund could be on it if he backs up one game in the Finals, if they consider backing up playing (which they should, IMO). And Boucher could be due to circumstances even if he doesn't dress.
I really think players who are on the roster the entire season even if merely depth, such as Cote, should have their names on it. Riley is a good guy in the room and works hard at practice even though he knows he isn't playing -- he sweats like a son of a gun at the SkateZone -- He wouldn't be carried if they thought he wasn't needed, and was serving a purpose.
I just hope this becomes a concern in a couple weeks.