You can really give yourself a headache trying to make sense out of caliber designations. It is not that there is no system, it is that there are several and you have to know which you are looking at. Not helped that Internet Usage is kind of sloppy and many posts say "bore" where they should have "groove." Plus the best and worst efforts of the advertising agencies. Best to just learn most of them by rote.
Horrible example, .257 Roberts has that groove/bullet diameter but the .256 Newton has that bore diameter and shoots .264" bullets like the 6.5mms.
As to the OP, if he wants a true 6mm bullet, there are some out there. The first I read of was known as the "23 and a half" = .235, but I think most of the ones left are .234"; which is not quite 6mm but is as close as you are going to get. Not inexpensive, though. Cheap requires a .243 Win.