One of the marks of a true professional is using proper terminology. Many nail technicians use these term incorrectly - including some big name companies that I know! A true professional should know what they are dealing with and they should be able to use the correct terms when talking to their clients.
I found this picture online and love how it shows what is the true cuticle vs what is the eponychium. True cuticle is dead tissue and there is absolutely no reason you cannot remove it - and if you are applying enhancements you MUST remove it or the nail will lift! The eponychium on the other hand is living tissue and should not be removed - it is sealing your matrix from infection.
Now, by definition, pterygium forms if there is scar tissue in the nail matrix. this is scar tissue and is NOT removable. I have attached a picture of this disorder. As you can see, you will not be able to remove this scar tissue and should not try. You cannot put an enhancement over pterygium as it is living tissue - the enhancement will not stick and it could lead to an eventual allergic reaction.