White Miller as a common name was originally applied to a wet fly, not an insect. It goes back very far. It had been in existence for some time by the 1890's when the fly by that name was suggested by Mary Orvis Marbury as good to use "when the white moths were about." People call things lots of names and it has been applied to the mayfly genus Ephoron by a few, but it has mostly been used by eastern anglers for the caddis genus Nectopsyche at least as far back as the early 20th century. The most common species is transcontinental and there are others as well. The name was applied to western populations only recently, after anglers became aware of them and they were confirmed to be the same critters as their eastern counterparts.
"It's not that I find fishing so important, it's just that I find all other endeavors of Man equally unimportant... And not nearly as much fun!" Robert Traver, Anatomy of a Fisherman
I remember reading about that pattern in Bergmans Trout. As I recall, he held high regard for that fly, especially for fishing for Browns. Not sure what species of mayfly it was supposed to represent and maybe it was more of an attractor pattern, but he swore by it,especially when fishing up in the catskills , as one of his real go-to patterns.