Will it still be there if a player is trying a tutorial game? ..I'm imagining a scenario of someone, say, showing a friend ADOM, and starting them a tutoral game on a PC on which ADOM has been played many times before. That cave teaches some very good points about the game, so I think it would be nice to have in the tutorial. But yes, very quickly get rid of it in the main game.

...and now that I've thought about it some more, the purpose of this dungeon is really as the tutorial dungeon, yes? Should it not just be a dungeon that appears when a tutorial game is started? And it's a randomised dungeon otherwise?

ie: if {s} start a tutorial game is selected, the starter dungeon is in the tutorial format, if {g} generate a new character is selected, the player is assumed to know what they're doing and it becomes a randomised dungeon.

Having a game-count before a permanent change seems to me to be the wrong metric, especially if someone is going to try to introduce a friend to ADOM on their own PC, rather than installing it on the friends PC. If the friend doesn't get to try the starter dungeon just because someone has already played ADOM many times on that PC, the tutorial is much less effective, yes?

I say base it on the game type chosen. If you still want a game-count for the standard {g} generate a new character game, that's fine. But I think it should be the tutorial dungeon every time a tutorial game is started.