I know that's what Larry Coon said in some chat, but his own CBAFAQ suggests Aminu falls under the Non Bird Exception that allows teams to resign their own Free Agents. I can't think of any reason why the players or the owners would agree to the limitation you cited for teams that aren't over the cap. Players wouldn't vote for something that limits their pay and owners wouldn't vote for something that limits their ability to keep their own players, so it just seems unfathomable that the rule would exist unless it was part of a salary cap exception.
NON-BIRD EXCEPTION -- This is also a component of the Veteran Free Agent exception. Its name is somewhat of a misnomer, since Non-Bird really is a form of Bird rights. Players who qualify for this exception are called "Non-Qualifying Veteran Free Agents" in the CBA. They are veteran free agents who are neither Qualifying Veteran Free Agents nor Early Qualifying Veteran Free Agents, and include the following:
Players who finished the season with a given team, who have played no more than one season without clearing waivers or changing teams as a free agent.
Players who were Early Bird free agents, but whose team renounced its right to use the Early Bird exception to re-sign the player.
Players who were to be Larry Bird or Early Bird free agents, were playing on one-year contracts, and were traded mid-season.
This exception allows a team to re-sign its own free agent to a salary starting at up to 120% of his salary in the previous season2 (not over the maximum salary, of course), 120% of the minimum salary, or the amount needed to tender a qualifying offer (if the player is a restricted free agent -- see question number 43), whichever is greater. Raises are limited to 4.5% of the salary in the first year of the contract, and contracts are limited to four seasons when this exception is used.
A partial season counts as a full season for the tenure calculation related to Bird rights. If a team signs another team's free agent to a rest-of-season contract mid-way through the season, then at the end of that season the player is a non-Bird free agent.
Starting January 10 of each season, this exception begins to reduce in value. See question number 26 for details.