"Should believers in Messiah be concerned with what fellow believers choose to call themselves such as, 'Messianic' or, 'Nazarene'?"
It's important for a group to come up with its own name (endonym) rather than allow an outsider to apply a name to them (exonym) because the latter name tends to be misleading. People tend to use exonyms to make their opponents look bad and use endonyms to make themselves look good.
For example, the label "One Law" is an exonym; the UMJC created this label to make Torah-observant Gentiles look like idiots, taking a "One Law" passage from the Tanak out of context and constructing an entire theology out of it.
For another example, consider Pro-Life or Pro-Choice. These are endonyms. I personally prefer to call "Pro-Choice" what it really is: murder. "Pro-Murdering Babies" would be an exonym for example.
Also, another concern with labels is that they are accurate. If they are accurate then they can be helpful for defining the group's identity and communicating that identity to others. If not, they can be harmful.
I hope that answers the question! : )
If anyone has anything else to add then please feel free.