I think you both have it wrong.
I think if the Isles put him back on waivers within a certain time frame (likely a month or less after first claiming him) then
only the teams which put in a claim for him when LA waived him can make a claim again. If none of these teams make a claim, or if there was no other team to make a claim, then LA could reclaim him if they so choose and send him straight to the Islanders.
I believe this is how it worked in the last CBA, but I could be wrong.
Of note, other teams claiming him could even impact if the Islanders want to trade him later, barring any changes to this issue under the new CBA. When the Islanders lost Jason Weimer (flashback moment there) on waivers to the Minnesota Wild, the Wild couldn't trade Weimer without dealing offering him to one of the other two teams to put in a claim on him (ottawa and the Rangers).
http://www.rotoworld.com/recent/nhl/457/jason-wiemer
If this rule is still in place (and it might be gone with two new CBA's in place since then) then the whole waiver claims process would impact on every form for which a player could be removed from the team that claimed him, and not just if he was placed back on waivers. In essence, you claim him, you better keep him, or you'll lose him for nothing.