Well yes. I know that the OoT Master Sword doesn't have the green straps, but I could have sworn TP Master Sword did... Hmmm... I need to play that again.
Correct. TP and SS both have green straps. However, TP has small bits of gold going down the ricasso ( the thin bit before the flare close to the guard before the Triforce symbol), and all in all looks more ornate.