It's a difficult thing to fully characterize 'Banshee'--particularly since it seems likely not to be just a single clone (although perhaps one or two seem to dominate). However, the different clones identified as fundamentally similar (foliage, prickles, growth, hardiness, receptacles, etc) have not been named in conjunction with their distinctions (usually having only to do with flower shape, color, and petal count, which in turn affects petal arrangement and balling), so there is no current way to handle their nomenclature on a clonal level. If you have read not read Leonie Bell's article in the 1977 American Rose Annual (also mentioned in a comment below), that is about as clear of a picture of the overall complexity as you will find anywhere.

I do not think that R. x suionum (misapplied, it would seem--see below) is distinct from 'Banshee' sensu lato; they are the same, and the button-eyed morph appears now and again just about everywhere under all of the various names we know 'Banshee' under. The only questions are as follows: Are plants grown as R. x suionum actually assignable to the original taxon that Sigfrid Almquist described (setting aside questions about the validity of that name for a moment)? One glance at the description is all that I need to say, unequivocally, "no"!

Here is the precise wording in the protologue for Rosa suionum: "R. (Gall. gl.) suionum AT. n. spec.: in Suicia media antiquitus culta cum R. francofurtana, corolla simplici alba, foliolorum forma et serratura omnio typi; hybriditatem vetant chromosomata certeraque omnia."

I am still figuring out some of that last bit, but here are a few key points:
1. No specimens were cited, nor any type specimen designated; now, if Almquist deposited any herbarium specimens anywhere, those would be very interesting to locate.
2. The rose he described was nothing like our modern 'Banshee', but rather, had simple (presumably five-petaled), white flowers. Almquist confirms this in the following article in the same journal (see below).
3. Most fascinatingly, this Rosa x suionum was growing together with R. x francofurtana, which is remarkably similar in certain characteristic features to 'Banshee', even though it does not seem likely to be at all the same plant. One wonders if its mention in the protologue might not have had some impact on the confusion that followed.

Here are a few links to check out:

The protologue of Rosa x suionum S. Almquist in Arkiv för botanik 16:23. 1921: https ://www.biodiversitylibrary.org/page/28797632 [this link needs to be pasted together with the space removed; I couldn't get it to post if I left it as an intact string, sorry.]

Almquist describes a bit more clearly the location, and repeats the description of R. x suionum that is absolutely nothing like 'Banshee': https ://www.biodiversitylibrary.org/page/28797680 [this link needs to be pasted together with the space removed.]

The story of how 'Minette' came to be the new name for these plants in Scandinavia, courtesy of author Hellmut Merker (who convinced himself and everyone else that Vibert's alba must be the same variety) and Hugo Lykke (who may have been responsible for the original mislabeling), in Merker, H. 1984: The Nordic Rose (Rosa suionum) unmasked. [The identification of Rosa suionum S. Almquist.] Swedish Bot. Tidskr. 78: 309-312: http ://www.diva-portal.org/smash/get/diva2:1202466/FULLTEXT01.pdf [I have managed to OCR the text and translate it to English, as well, if anyone is interested. This link also needs to be pasted together with the space removed.]

What is really needed, ultimately, is a massive DNA study that includes many species and cultivars of good provenance, all vouchered, and hopefully represented by multiple accessions each.

More later, perhaps, but that should give everyone a bit more to chew on!