You are correct, they use the same type of tube. Fat neck.
It's really tough to make a call on what brands/models of thrift store TV's to watch out for, but I can tell you this: if you can see inside the back of the TV, you can usually tell if it's a fat or skinny neck.
You can also tell if the yoke is bonded (glued on) or not.
Also, look to see if you can tell if the tube's mounting ears are mounted forward or backward. You want a tube with the ears mounted toward the BACK of the tube, not forward. If you swap in a tube with the mounting tabs facing forward, you'll run the risk of your neck board not clearing the back door, and potentially have issues with a molded tube bezel not fitting properly.
You CAN insert spacers or blocks of wood between the tube's mounting ears and the frame, but it doesn't always work properly, depending on the game your swapping tubes for.