[dumbo-octopus]

They're related, but in a very specific way that does not meet the bar for "variation" in my opinion. As you described, counting sort is a very simple procedure that works well only on a very specific input domain. If another algorithm has "more logic" than counting sort itself in order to transform a very different input domain into a format that is suited for making many repeated calls to (a procedure that may be implemented as) counting sort, in a specific pattern, and appropriately coalescing the results of those calls, I think it is appropriate to let it have its own name. Would you prefer to refer to every possible utilization of counting as "that version of counting sort where...."?