The protostome/deuterostome distinction and why it is important
The terminology is really just describing the process in embryonic development that you describe: deuterostome means "second mouth"; protostome means "first mouth."

The significance of these groups is in evolutionary history and cladistics. All of the deuterostomes have a shared common ancestor that is not shared with any protostomes, and vice versa. Regardless of developmental pattern or number of openings during development, no other organism should be given these names (and if it is, and the correct ancestry is determined later, they would be moved out of those categories accordingly).

This is actually really important in biological classification of organisms, and before modern genetic techniques these types of shared characteristics were the best way to determine shared ancestry. Think a bit about what it means for an organism with two openings to differentiate one as mouth and the other as anus: once they are strongly differentiated, there is a huge barrier towards going "backward" and "switching" openings because mouth and anus have very different functions. However, if you go from having one or no openings to having two, before there is specialization, the distinction of which one develops first isn't likely to be that advantageous one way or the other.

Nephrozoa contains the deuterostomes and protostomes, consisting of almost all extant animals. However there are also other animals that are neither. For example, Xenacoelomorpha might be the closest thing to a bilateral animal with only one orifice, though they don't really have a true gut at all.