I first learned thumb and two finger Piedmont style, but also do thumb and index as well, depending on what I’m playing - sometimes that’s the most natural way to get the feel right. Some other famous thumb and one finger players include Magic Sam, Doc Watson (though he was mostly known for flat-picking), Reverend Gary Davis, Wild Jimmy Spruill, and lots of others.
Elizabeth Cotten had a “strange” style because she was left-handed, and learned to play on right-handed guitars, so she learned to play upside down. Her index finger did the thumb part, and the thumb did the index finger part. Weird or not, I love listening to and playing her stuff (though I play it right-handed, and usually use thumb and two fingers, just because I figured out how to play her stuff by listening, so did it using the techniques I was most comfortable with.)
I don’t know that I’d call these styles “weird” - they’re traditional country blues styles. There are a number of distinct styles, all different, though some players are hard to put into one of the bins. (Like Mississippi John Hurt, who was billed as a Delta bluesman, though his style wasn’t remotely Delta. He used thumb and two fingers.) I think it was Stefan Grossman who said that you can’t get a real blues feel if you use thumb and three fingers, “fingerstyle”.
In fact, modern thumb and three finger “fingerstyle” is only one of many flavors of fingerpicking, and not common in traditional American folk and blues styles. So maybe it’s the weird one? Or maybe it’s just all good...
