Does having this feature depend on how particular carriers provisioned their phones? I'm not clear on how to do this with my T-Mobile S7 Edge.
It depends on which way you are trying to use digits.
If you are trying to have one number on multiple devices it is handled on T-Mobiles side not handled through your phone so I think that is why you do not need the app. So you should not have to do anything after everything is set up on the T-Mo side. I did have to give them my watch information when setting things up so it would work correctly.
If you are trying to have multiple numbers on your S7 Edge then yes you need the app.
I would assume most people in the S3 area of the forum want their watches to send out their phones number (one number on multiple devices) so they do not have to give out the watches number out and confuse people. Since I have been a beta user on digits my watch does not even use data according to my account. Prior to using digits it did because I do leave my phone pretty regularly at work and when I workout in the mornings.
If you think you configured it incorrectly you can re-register with digits but again you do NOT need the app on your phone if you are trying to do the one number on multiple devices.
Hope this does not confuse you more than it helps.