I think nictitating membrane is the word you are looking for.Bad times but that still made me laugh
To be honest I didn't know cats had a 3rd eyelid till last night.
A proper name does exist but we all know I'm just looking at google, he knows everything.
I had a dog that had that done so her cornea could heal. She looked really creepy with the red eye for a while, plus she got to wear the cone of shame while we waited for everything to heal up.
Bad times but that still made me laugh