It's not necessarily the primer itself. Not sure if the mineral spirits would change things, very well could have, but in my opinion the cause of your mishap is that the tape should have been removed immediately after you painted before the paint was dry. If you have tape anywhere else you've not yet removed, score a line along the edge of the paint before attempting to remove the tape.

I've used Zinsser "Guardz" primer with much success, it's claim to fame is sealing damaged drywall--FWIW I've used it over messed up wallpaper removal where it's impossible to get all the glue off. Check it out. Whatever you do I'd advise a small test patch before proceeding.

Basements are damp, and your paint simply isn't drying. You would most likely have had this problem even with the best brand of paints. Removing the tape immediately and slowly is your best defense, but in these situations if you can paint the room without it, that's the way to go. If you must use tape, try the green one instead of blue.
If the primer is water-based, not sure what the negative effects of having mineral spirits mixed in it. I would think you would have had difficulty in the application process, but that's never happened to me, so don't know.
To repair those areas that peeled, I would use wallboard compound, sand and prime. Bullseye 123 is fine. If it's new, unfinished drywall, I would use a PVA primer.

I found that I have to remove the painters tape before the wall begins to dry. Now, I just don't use the painters tape. For me, it's just easier to not tape and spend extra time doing it without it.

I don't usually use painter's tape but that certainly COULD be the problem. I used it this time because I was painting an accent wall a darker color than the adjacent walls and the lines needed to be perfect or it would be glaring. I may paint the closet in my son's room and see if I have any issues.

While I know MOST basements are damp, I live in ultra-dry Utah where we have single digit humidity year round (if any). The paint is dry in this case though I know that can be a problem.

Is my method of fixing my mistake okay? Should I re-prime the unpainted room or not?

I'm trying to remember what I did when this happened to me when our daughter and I were painting our new home seven years ago we made the mistake of leaving paint on in one room and this happened to us. I think what I did was carefully remove the loose paint, sand the edges lightly in hopes of removing any signs of the mishap the repaint.
I've been in lots of pain this past week which causes memory lapses so I can't guarantee I'm remembering this correctly. It should work though, especially if you prime after sanding. Good luck!