I may be wrong but I thank that if a guy goes through waivers and is picked up by another team and then waived, he goes to the minor league team of the original team automatically unless someone behind the team that claimed him picks him up.
Not quite
If player is claimed (by Team B) and then but back onto waivers, a normal waiver process begins.
The player goes to the claiming team with highest priority. That may be the original team (Team A), or it may be a previously involved third team (Team C).
If a Team C successfully claims him, then they are in the same boat as Team B was - play him in the NHL or put him through the normal waiver process.
If the original Team A successfully claims him there are two possible outcomes:
1) Team A was the successful claiming team, but there were other teams also putting in a claim. In this case Team A is subject to the same conditions as Team C would have been and as Team B was previously - play him in the NHL or put him through the normal waiver process
2) Team A is the
only claiming team. In this case Team A
can lend the player to an affiliate league without waivers, however it is not automatic and they can choose to keep him in the NHL if they choose.
The idea behind waivers is to keep a player in the NHL.
If Team A thinks a player is "not NHL worthy" they can send him down through waivers and offer up that player to anyone else who thinks he is NHL worthy.
If he clears, that in effect, is the whole League saying he is not NHL worthy.
If he is claimed, that's someone saying "Hey, wait up, we think he is an NHL player". He passes muster, remains an NHL player, and any worthiness-check needs to be made again if his new team wants to send him down.
If the new team realizes, okay maybe he isn't an NHL player and want to waive him, three possible checks are made.
1) He clears waivers - no one thinks he is an NHL player. He goes down.
2) A third team puts in a claim. Whether successful or unsuccessful, it doesn't matter. All that matters is there is someone in the League saying, "we still think this guy has potential to play on our NHL roster". He remains an NHL players.
3) No one puts in a claim other than the original team - original team can effectively say, "see we told you so, and since we already declared him as not NHL worthy, and you, his new team, also don't think he's NHL worthy, and no one else in the League is speaking up, then it's as if he passed waivers, and if we want to send him down now, we're just gonna do so without asking permission again".
Or for a third analogy, and one that is my favourite...
You're at a party and ask, "hey, does anyone mind if I finish this bowl of chips?"
If there is no protest, you finish them.
If somebody wants some, you pass the bowl along.
But then imagine that new person has their fill, and decides they are done, and they ask "hey, does anyone want the rest of these chips?"
If someone else this time wants some, you share with them or let them finish it.
Bu if no one else says anything and every one has had their fair chance to eat some chips, and they don't want to, you have fair game to just finish the bowl yourself.