OK first Unlocked means unbranded, so if it says T-Mo it is a T-Mo branded phone. Many of the carriers do not permanently lock devices anymore. As an example Verizon devices usually auto unlock after 60 days. It is still a Verizon branded phone, meaning it has all of their bloatware and added junk on it but you can take it to almost any other carrier and have it activated. Some times there are still compatibility issues. I've seen several instances where Verizon branded devices taken to AT&T have messaging issues. It requires AT&T to do something on their system to get it to work and some times it requires software Flashing.
Long story short if you want an unlocked device get an unlocked device. If you want T-Mo's bloat and extras that may or may not work, or may or may not cause you issues on another carrier get the T-Mo device.