i might be misunderstanding the question, but there's no rules about a waiver claim having to stay with their new nhl team. it's just that they have to go through waivers again to be sent to the ahl.
EDIT 2: I mis-understood the question and response. Your response was about a guy claimed, and there is NO days/games played associated with that. A team claiming a guy would need to put him on waivers again to send him down, just as you stated.
My response below was in regards to a guy not claimed, as I misread the posts.
Isn't that only if they are on the roster for so many days and/or play in so many games?
EDIT: 13.5 of the CBA says 30 days or 10 games.
So a player not "loaned" to a minor league club or one who gets recalled from a minor league club won't need to be waived as long as he is under 30 days or 10 games, both cumulative. I don't think pre-season games count, but it's not clear.