This is actually a trademark issue and not a copyright issue. If someone were copying your songs, that would be a copyright issue. If someone is using the same name as you, it is a trademark issue.
That said, trademark ownership and infringement is an extremely fact-specific question. Someone needs to know at least (1) was he really using the name first, (2) if so, how large a geographic area he was using it in, (3) how large a geographic area you are using it in, (4) whether either person has filed an application for federal registration, and probably several other things that come up based on the answers to these questions.
Bottom line, if the other artist was already using the name, he likely has superior rights to you. They may be limited to a specific geography, but maybe not.
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"Who has rights to the name?" In what sense? It is unlikely you two have a cause of action against each other which is worth pursuing in court.
If you want file a state or federal trademark (not copyright) claim, that will be a long and expensive process which may accomplish nothing other than wasting time and money. The internet complicates everything today, and things like trademark are catching up slowly. Courts are very slow to act.
This may sound bizarre but a few performance artist groups, like clowns and magicians, register things related to their trade and police their own members. If there is an organization that registers DJ names then you could possibly go to that organization and have them mediate the dispute. I don't know if an organization like that exists, or how effective it would be, you would know better than I since you are in the industry.
I would contact an entertainment attorney in your area and discuss all the facts to see what can be done.
This is a factual question and the outcome will be determined by the facts. This is also a trademark issue regarding the use of the mark DJ DISTINCT. Trademark rights are created through usage in a particular territory and trademark infringement is based on confusion/dilution. These trademark rights are created via a first-in-time argument where the first person to use the mark in the territory is considered the owner of the mark.
Some questions to ask the IP attorney you retain:
1. Do you know if the other artist registered any aspects of the DJ DISTINCT mark with the USPTO or with any specific States?
2. If yes, then what date did the artist claim as his date of first use in interstate commerce?
3. Is this date before 2010?
4. Do you know if the other artist created any DJ DISTINCT logos for usage in commercial or promotional items prior to 2010?
5. If so, were any of these DJ DISTINCT logos copyrighted prior to 2010?
If there are no active registrations or 1B applications on file, then the matter will come down to proof of the alleged facts (i.e., each side will have to prove its usage based on the admissible evidence). If the other artist has indeed been using the name since 2006, then he will have to prove that his usage pre-dates your usage in the applicable classes of goods.
The foregoing response is provided for general informational purposes only and is not a solicitation for business. Please retain an attorney if you need specific legal advice. No attorney-client relationship is established until both you and me agree to establish one, and neither transmission of information herein, nor the receipt of such information, constitutes an agreement to establish an attorney-client relationship.