I cannot opine on the Lynnette/Paula/Zach love triangle, as I had to Google those names to even know what you were talking about.
Yeah, people throw the term "ring chaser" around. It helps to clarify, so I'll clarify further (as this is a more interesting discussion at this point than the Ray stuff, which is going in circles).
I think the worst example you can point to is Karl Malone and Gary Payton joining the Lakers. Two vets, who had led teams to the Finals but lost, jumped onto a team they were CERTAIN would win a title for them. That hits all my points of ring chasing: (a) a past-his-prime superstar who (b) hasn't won a title and (c) has tons of cash and who (d) foregoes more short-term money and his long-time team to (e) sign with a team he expects to win him a ring (e) without him having to do much besides come off the bench and hit a few shots.
I was offended by that. Fans are offended by it. (If a player is traded, that's different. For as harsh as Boston fans can be, recall how supportive they were of Ray Borque when he won a cup with the Avalanche.) And at least to me, it matters not one whit whether or not the team actually wins a title or not. If you chased a ring, you chased it, regardless of whether you got it or not.
So the Malone/Payton situation is the perfect example. When a player signs on expecting to play a major role, I think it's a little different. Take Rasheed Wallace on the Celtics. (They needed him to play major minutes -- unfortunately, neither they nor he expected him to play MAJOR minutes in Perkins's absence from the lineup.) Yeah, he joined a great team, but that team needed him. (Plus he had won a title.)
So, returning to Ray for a minute, I don't think this a bad case of ring-chasing. It seems clear that he left not because he thought Miami was better but, rather, because he wanted out of Boston. Would he have signed with some random team if Miami wasn't interested? Probably not. He probably would have swallowed his medicine and re-signed. But this opportunity presented itself and he took it. I don't think it's to burnish his legacy; I don't really see how it could. He already has a title; being a supporting player on a couple more won't really change what people think about how good a player he was. The only thing for sure is that #20 won't be hanging in the Garden rafters, which, as a UConn and Celtics fan, is a real shame.