Text styles are saved with the Project. Look in the Project Panel. The only way I see to delete a style is in the Project Panel: Right click and you can "clear" (delete) the style.
You can also right click and Export the text style (a .prtextstyle extension).
And in the Project Panel, right click -> Import - to get the style into another project.
You can also open both projects, copy the style item in one Project Panel, and then paste it into another. But even with both projects open, you only see the styles for the project of the EGP item you are working with. Ah, but not true! With a sequence from Project 1 on the timeline, I can look in the Project Panel for Project 2, and drag a style from that project onto the Project 1 graphic. And that adds the text style to Project 1, where it is now available in the Style drop down list.
Caption "Track Styles" and "Styles" appear to be the same.
If you rename the style file outside PR, it is an independent style, but imports with the original name. You can change the name once in PR to something different.
I do not see a way to "save" a style once it is updated. It will still sync back to the original style. So you have to create a new style with a new name.