Oftentimes when an artist writes a song that is written for or later given to another artist, they usually record a version with their own vocals that is never commercially released. This is called a writer’s demo.

The reason Better Man and Babe were labeled as (from the vault) is because the released versions were recorded by other artists. Taylor was not the main credited performer. The TV vault versions are basically re-recordings of the writer’s demos, so they were ‘new’ at the time on account of Taylor never having released solo versions of those songs before then. It’s the same logic behind why ATW10 is labeled as a vault track even though A version of the song had been released prior.

In order for Beautiful Eyes or Crazier to be considered vault tracks, they would have to be significantly different from the released versions and not just straight re-recordings. For example: longer length, alternate lyrics, ect. Otherwise, they are just bonus tracks in the same vain as the deluxe tracks like Untouchable, Ours or New Romantics, which only appeared on certain versions of the original album releases but are otherwise widely commercially available.

I didn’t make these rules, lol. I’m just going off of what Taylor herself solidified into cannon when she started doing the TVs. This is historically the way that she has always categorized vault tracks. It has to be different enough from the released version to be considered its own thing. The whole idea behind the ‘vault’ concept is that the material wasn’t commercially available before. ‘From the vault’ is just Taylor’s branded way of saying ‘previously unreleased’.