"Looks like it could be this plant
Joseph's Coat (Opuntia monacanthos 'Variegata')"
Wow, I do believe you are right! Thank you so much ! When looking up Cacti, that particular plant never came up, but once I looked up the name, I see lots about it. Honestly I've been surprised it's lasted so long as well considering I drop a dollop of water in it when it's dry and have left it in that shallow pot forever without doing anything else besides putting it outside on the deck with the dr acaenas. I have seen it turn a little pink sometimes, but thought that perhaps it was getting sunburned and brought it back inside. Hopefully it likes the new cactus soil, although I added a bunch of pumice to what was in the bag because the 'cactus' soil seemed to look/feel way too much like regular plant soil for my tastes.
Question about all the little shoots it puts out every year: should I pinch some of them off, so the plant can concentrate on just some of them? Or should I just leave them alone and just pinch them after they shrivel up?