Characters in a string can be thought of as digits in a base-256 number. Call counting sort recursively on each bucket, looking at the next character in the string. Can you not see the similarity?
There is no one who fails to see the similarity, because they're both kinds of radix sorts. For instance, a counting sort is a degenerate form of radix sort that only does one pass.
But where the classic radix sort is bottom-up, the American flag sort is top-down. That really matters for data where the the most significant portions of the data is likely to be all you need to consider for the sorting. While a top-down sort will have similar performance to bottom-up in the worst case, in the best case it can skip a lot of passes, especially if the inputs are much longer than the longest common prefix.
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...."?
But where the classic radix sort is bottom-up, the American flag sort is top-down. That really matters for data where the the most significant portions of the data is likely to be all you need to consider for the sorting. While a top-down sort will have similar performance to bottom-up in the worst case, in the best case it can skip a lot of passes, especially if the inputs are much longer than the longest common prefix.