"[Battier is] as smart as hell. He understands how to play the game. He plays it at 100% ferocity all the time, a ferocity with sense. And on top of that, he's a character guy, he's a leader, he's a consummate pro. I hate him." - Gregg Popovich
Miami did not renounce Wade because they wanted to keep him (obviously), and they didn't renounce Anthony because his cap hold was so small. After Anthony opted out, Miami extended him a qualifying offer, which they were perfectly capable of doing since he had only three years experience. You can make any free agents of yours with three years or less experience into restricted free agents, whether they like it or not, by extending a qualifying offer. The only exception is players on rookie scale contracts who had options declined, but this did not apply here. (Kenny Hasbrouck was also eligible for a qualifying offer, but did not get one.) Anthony's QO was equal to the minimum salary plus $175,000 (a total of $1,029,389), and his cap hold was equal to his qualifying offer.
[...] Joel Anthony's contract did not require cap space. Because Anthony had spent three years with the Heat without changing teams as a free agent, he was a "qualifying veteran free agent" - a lawyerish way of saying they had full Bird rights on him. As such, they could re-sign Anthony for whatever they wanted using the Larry Bird exception, and did so when they re-signed him to a five year, $18,250,000 contract after the Haslem deal was closed. His cap hold was incredibly small, and he's BYC because of the big jump in his salary, yet there was nothing untoward behind how they were able to sign him. It was all a very well planned and timed case of creative financing. (Why they gave $18 million to a 28 year old who averages 2/2 and who is both a poor scorer and rebounder is not immediately obvious - his defense isn't THAT good - yet we'll overlook that for now.)
Joel Anthony - This is the only time I will ever campaign for the signing of a 28 year old 7 footer who can neither score nor rebound. Joel Anthony is an effective interior defender who blocks and changes shots in ways best not measured. There is nothing else about his game to like, but as a third stringer, there doesn't need to be either.
Note: Non-US teams that the player
has played for are, unless stated otherwise, from the top division in
that nation. If a league or division name is expressly stated, it's not
the top division. The only exceptions to this are the rare occasions where
no one league is said to be above the other, such as with the JBL/BJ League
split in Japan.