(Except for whatever goofy thing H.N. White used on the Trombonium)
Now what? I don't have horns to test all of them on because I don't have anything bigger than medium shank (yet).
There have been at least two tapers. 5% (standard) and whatever Conn and Besson were using. I can't for the life of me get a consistent measurement to figure it out and it's too close to figure out where the wobble comes from. Maybe it's 4%. Maybe it's 6%. I dunno. I don't have a horn with a big enough receiver to confirm that the two smaller Conn "Remington" tapers are the same pitch as Remington Bass Trombone shank.
Small shank is small shank. It's not very interesting.
There is a not so well documented Conn small shank (11mm). Probably not so well documented because it works either way if you ignore the wobble. If you insert a Besson medium shank (or whatever) mouthpiece into the receiver of an old style Conn 20-I, it will stick out a mile, but it won't wobble.
Medium shank is an (in)appropriate description of like 4 different shanks. Even better that the insertion depth is just as consistent. (ie it isn't)
Conn medium shank (11.5mm) seems like it should be compatible with "Euro" shank (11.7-12mm?). "Bariton shaft" (11.5mm @ 5%) and Russian Baritone shank (12mm @ 5%) aren't the same thing. Insertion depth with these is all over the place, also. My Soviet horns are designed for a whopping 20mm depth, so the mouthpieces look like they're hanging out a mile. If you insert a "Euro" shank mouthpiece into a 12mm shank reciever, it works alright with some wobble. If you insert a "Euro" shank mouthpiece into a 11.5mm shank receiver, something is getting damaged. I bought a Besson 10 hoping to get a more official measurement, but it has too much damage to teach me anything. Not a bad mouthpiece though!
The Tuba mouthpiece in the photo is from a B&F stencil Eb Bass. It's using the proprietary taper medium shank for some reason. Such a huge throat and small shank is not a good combination for intonation! Again, I can't measure it well enough to get a taper measurement. It seems to only insert 20mm and the receiver is about 13mm. So if I try to calculate the taper...12mm = 5% (wrong), 11.9 = 5.5%, 11.8 = 6%, 11.7 = 6.5%. See the problem?
Large shank is all over the place. I see why adjustable gap receivers are a thing. According to Josef Klier, whom I trust more than most, large shank should be 12.5mm and the "other" Bass Trombone shank is 12.65mm. Bach lists large shank as ~12.6mm (.496"). A large shank Kelly 51D measures 12.5mm if you catch it just right. I have an unmarked mouthpiece with a shank that is clearly enlarged (slightly) from another template and it measures 12.65mm. And then I got some random Chinese mouthpiece made on a Denis Wick blank which measures 12.75mm! It's a lot shorter than it should be and the backbore is huge, so that accounts for it a little bit, but...yeah. 12.75 to 12.5 is 5mm in shank length!
Remington shank (13mm) is enormous. I have a Remington 6-1/2AL (why is this a thing?) and it won't even catch on anything that I own, so I can't confirm that this is the same taper as the other Conn shank sizes. Supposedly, they're related.