A person can use any name they want to use, as long as it isn't for fraudulent purposes.
For instance, unless it's a formal setting, I usually go by "Bob" instead of "Robert."
If David Joseph Miller and David Joe Miller are the same person, the discrepancy could have occurred in any of a number of places along with road. Without following the paper trial, it is impossible to know where.
I agree with the assessments of Mr. Marshall. The difference of names could have been the result of too numerous an event to guess where the differnet names became entwined. Essentially, the difference is not significant enough, unless there were two persons with one using one name and the other person using the second name. Good luck.
Since I do not practice law in your State, this answer is provided solely for informational purposes only, for you to use as a starting point when speaking directly with a lawyer in your State. This answer does not constitute legal advice, create an attorney-client relationship, or constitute attorney advertising. I urge you to immediately contact an experienced criminal defense lawyer admitted to practice law in your State before you make any decisions about this case.
I would also like to add that many criminal defendants have multiple aliases associated with their name, all of which are tied to their name in their court record. Having multiple names isn't necessarily a "mistake" on anybody's part. If a person's ID at the time of arrest stated David Joe Miller, but their death certificate states David Joseph Miller, that's not necessarily a mistake on anybody's part.