Phrag. schlimii 'Wilcox' is a vigorous clone that multiplies quickly. It is also a VERY beautiful clone that has stood the test of time and it remains popular in collections everywhere. However, as everyone knows, it is actually a Phrag. Cardinale. The judges who awarded it as a schlimii were asleep at the wheel.....and they have been the laughing stock of the AOS award system ever since.
However, by any name, this is a spectacular plant and it has been spread around a lot. Occasionally, a grower has renamed his piece with the correct grex name; Cardinale.....and resubmitted it for judging, along with a new clonal name. As a result, a number of pieces of "schlimii 'Willcox' " have been reawarded as Cardinale. All of the following Cardinale's are actually pieces of the original schlimii 'Wilcox':
Cardinale 'Birchwood' AM/AOS
Cardinale 'Westonbirt' AM/AOS
Cardinale 'Kilworth' AM/AOS
There are probably more; but, these 3 (especially 'Birchwood' and 'Westonbirt'), are the most common.
Personally, whether you call it 'Birchwood', 'Westonbirt', or 'Kilworth' doesn't matter to me. I'm just glad that it's got it's correct grex name now: Cardinale. Anyone who insists on leaving the bogus name "schlimii" on their plant is wilfully perpetuating the original fraud and that's just wrong.
I agree, Michel, that your plant probably originated from the 'Birchwood' clone which was first awarded in 1970. However, Dr. Dick Clements gave it the correct name - Cardinale; changing it from the incorrect name - schlimii 'Wilcox'.....and he added his own clonal name 'Birchwood', a name he often gave to his best plants. In the case of your plant, I'd leave the name as it is; because it is correct. It is a Phrag. Cardinale and it likely did originate from the piece that Dr. Clements grew and had awarded - 'Birchwood'. So, the provinence of your plant includes the fact that it was once part of the Clements' collection and since it was awarded with the correct grex name, it's all good!