AFAIK they still retain their original names.

You have the Amino group, Carboxyl Group which are alternatively referred to as terminus (N-terminus, C-terminus) if they are at the first and last amino acid in the chain respectively.

Then you have the R group that is also attached to the alpha-carbon of the amino acid. There is a fourth bond on this alpha carbon to a hydrogen.

To answer your question the carboxyl group from the newly added amino acid would (temporarily) be the c-terminus until a new amino acid is added to extend the chain.

If there are other names (I’m sure there are) they aren’t as commonly used.