Major labels are reportedly “overhauling contracts” for newly signed artists, with some now restricting artists from re-recording their music no earlier than 10 years or more after leaving the label.
This video is basically an ad for a new venture but it explains rights and payment well.
https://youtu.be/-TM9foEQJ-Q?si=JekKRS0pyFs9SC7B
EDIT: I believe that Taylor already owned her catalog but she didn't own her masters (the actual recordings). Therefore she was able to re-record her albums because she owns her songs, just not the original recordings.
There's also publishing.
This video is basically an ad for a new venture but it explains rights and payment well.
https://youtu.be/-TM9foEQJ-Q?si=JekKRS0pyFs9SC7B
EDIT: I believe that Taylor already owned her catalog but she didn't own her masters (the actual recordings). Therefore she was able to re-record her albums because she owns her songs, just not the original recordings.
I found a YouTube link in your comment. Here are links to the same video on alternative frontends that protect your privacy: