I don't know that there are any formal definitions, but I think it would be reasonable to call a 10 to 11 inch diameter satin-finish reflector a medium-size studio reflector.
The sort of reflector you'd use on a flashgun generally has a polished interior - without this, you wouldn't be able to focus the light for use at longer distances. In a studio, you're generally working at fairly close distances, so this is not too important. A satin finish gives a little softer light source. You can probably get one for your Profoto gear and modify it, if you think it's worth it.
I grew up with flashbulbs, even used two flashguns on rare occasions. But I wouldn't go back to 'em without good reason. (I guess that showing people it can be done would be good reason, but a couple of 12-packs would be enough for me.) Good luck in your endeavors.