I'm really looking into starting this as well, I'm not sure if it's taken off in MI, but I may even just do it for fun on my own property, but as far as the gun goes, regulation is you have to use a .45 revolver??
Yes, all three organizations have standardized on single action, .45LC revolvers. They also use black powder blanks.
Even if you plan to load your own blanks, I would suggest purchasing some 1/2 loads and full loads from a "certified" source. This will give you a consistent and safe load for training. It can also be used as a reference point for creating your own.
Note that .45s and the black powder both make for a unique sound. It is a low "boom." Some people have access to a .22 starter pistol (or even firecrackers!) and think those make a good training tool. While it can work, I disagree. Smaller calibers and smokeless powder both provide a much higher pitched "crack" sound. This, IMO, is more likely to irritate the horse.
When I shoot my .22 starter pistol, my ears ring. It is very uncomfortable. Horses have better hearing and I can only imagine how uncomfortable it is for them. Not the way you want to start training.
By contrast, my .45 with a full black powder load doesn't bother my ears at all. The low "boom" is much easier to tolerate.
If you have a SASS club in your area, they can often be a great source of information on black powder and reloads.
Hope this helps.