We have some more information on Cinderella's carriage thanks to someone who commented anonymously on the original post:
This is a rather typical Landau style of carriage. https://horseyhooves.com/types-of-horse-drawn-carriages/
Made by Les Voitures Robert Carriages, this is probably their Classic, though they also make them for ponies and miniature horses, Vis a Vis, they're one of the largest and one of the last remaining carriage makers in the world, it's a dying art form, sadly. They do actually make the true Cinderella style carriage which is essentially this carriage with a modified roof. http://voituresrobert.com/portfolio/classic-vis-a-vis/?lang=en
This one is set up for a single horse with shafts rather than a team with a central pole, and for some reason the top is not covering the rear facing seats, which it usually would. I am not sure what the knobs are, but would assume you're correct. The metal brake on the driver's foot well is hydraulic, a safety feature to prevent runaway.
Thanks for sharing! And if you'd like to let us know who you are, you can email me at horsiemama8@yahoo.com.

