Yes, Jeremy I got it now. Thanks.
The problem was, that when you said 'The obvious generalization of the rules of check is that the king is in check if it could be captured before the owner's next move'
I was thinking that, if in the first enemy turn, your king is not in check, but the enemy will be able to put your king in check; this would not count as putting your king into check, because the enemy is putting your into a check situation and not you.
Also rule 1.2 from f.i.d.e. chess handbook, talks about not being able to capture the king:
'[...]’capturing’ the opponent’s king are not allowed. The opponent whose king has been checkmated has lost the game.'
http://www.fide.com/fide/handbook.html?id=124&view=article