Terms used in Free Agency in the past have been incorrect and ambiguous. I'd like to see us to use the correct terminology and explain what each thing means. Some of our rules in the past have used ambiguous terminology and are not always correct (for example, we have used terms like "their own players" in the past when discussing who can receive 12.5% raises and that isn't correct as such raises ALSO require Bird Rights).
1. Bird Years
Listed on a player page, Bird Years count how many seasons a player has spent with a team without changing teams via free agency. When a player signs with a new team during the free agent period, his Bird Years are re-set to zero. Any other movement (e.g., trading a player) does not affect his Bird Years. At the end of each season, every player on a roster has his Bird Years number incremented by one.
2. Bird Rights
A player and a team have "Bird Rights" if (a) the player is a free agent, (b) the player last played with that team, and (c) the player has at least three Bird Years. A team may offer a player that holds Bird Rights with that team (a) any contract offer up the maximum based on that player's years of service in the league regardless of cap space, (b) a 5th year on a contract and (c) 12.5% raises instead of 10%.
For the purposes of Restricted Free Agency, matching a contract should NOT reset Bird Years. Sign-and-trades should also NOT reset Bird Years.
Basically, we need to expunge the term "your own free agent" (not specific enough) from our collective vocabulary and replace it with "a player your team has Bird Rights with" to accurately reflect the rules.
A couple of tactical notes:
Declining a rookie's 4th year contract option (after exercising his 3rd year option) makes him an Unrestricted Free Agent; however, the team will retain his Bird Rights and will be able to offer him a supermax after 3 years. (Why would you do this? To offer a smaller supermax than you would have to if you waited another year and the cap went up.)
Signing a player to a series of three one-year deals gets you Bird Rights with that player and allows you to offer him any amount you want. (Why would you want to do this? Maybe you find a young bench player that you get on a minimum for three years running, then offer him slightly more than the minimum (say, $100K more) on a long-term deal even though you're over the cap.)