Sorry but that is wrong. If an NFL player is not a "vested Vetran" (4 years on an active roster) when he is cut any team can pick up his contract and the player must go to whomever picks him. But if a player is "vested" then when he is cut, he becomes a free agent. There is a reason why young players are "waived" and older players are "cut". Now to claim a player the day after he is waived any teams interested put in a bid. If more than one team puts in a bid for a player then the team with the worst record last year receives the player. Therefore if any young player is cut, and we want them, we get them.
Now if a player is "waived" and then no one claims him then he does become a free agent, and ussually its also the end of his career.
Bottom line: Players with less than 4 years in the league have almost no control of their contract, and place they play.