AFAIK, this is exactly right. In order for Butler to get traded from the Patriots to the Saints, he has to be under contract with the Patriots. He is not currently under contract with the Patriots. He is a FREE AGENT at the moment. A *RESTRICTED* free agent, not an *UNRESTRICTED* one, and that difference, as Butler is discovering, is enormous. Gilmore was UNrestricted, meaning he could sign with any team, for whatever price he could get. Butler is free to negotiate a contract with any team, just like Gilmore was, but the moment he signs that contract offer, the Patriots have the last word - they can match or not match. If they match, Butler ends up in New England under those same exact terms he signed for with the other team. If they don't match, Butler ends up on that new team, but the Patriots, by rule, end up with that team's 1st round draft pick.
If Butler wants to be *TRADED* to another team, he first has to be under contract with the Patriots, so that means he has to either (a) sign the $3.9 million tender the Patriots offered him, or (b) negotiate another contract with the Patriots.
If he signs the $3.9 million tender, then the Patriots can THEN choose to trade him for whatever the best offer is they can get in return (maybe another team wants to give more than a 1st round pick; maybe the best they can get is a 3rd round pick...WHO KNOWS). OR they can simply choose to keep him on the team for 2017 at the $3.9 million contract they offered, and Butler signed.
Butler wonders why he isn't making as much money as Gilmore? This is why. He's a *restricted* free agent, which means there are all sorts of strings attached to teams looking to sign him. He's not an *unrestricted* free agent like Gilmore was.
People are trying to make this too complicated. It's actually not.