No. The very article you link details the difference, and that Counting Sort is often a subroutine in Radix Sort, and that Counting Sort is extremely poorly suited to strings, which is the entire purpose of this sort. And this name is better. So... "no" in basically every way possible.
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...."?
American flag sort is essentially an in-place version of counting sort.
That's talking about the single-pass American flag sort. If you use American flag sort to sort strings, you do multiple passes: on the first pass, you sort all strings by their first character, then for each starting character, you sort the strings with that starting character (which are now in a contiguous subarray) by their second character, and so on.
@chowells calls it a “top-down radix sort” below, which is a great description. It also explains the strengths and weaknesses of the two algorithms: radix sort works great for small strings of fixed length (e.g. IPv4 addresses, which can be thought of as 4-byte sequences) while American flag sort works great for variable-length strings like actual textual strings, especially if they don't share common prefixes (e.g. dictionary words, usernames, etc.)
@hinkley pointed out that the recursive version is just a bucket sort, but in-place. Which is also true!
tl;dr:
Single-pass American flag sort = in-place counting sort
Recursive American flag sort = in-place bucket sort