I guess I'll come right out and admit that I'm one of the crazy ones that has paid what we can all agree is a WAY above-market price for a MCO to WDW car service.... so I can help answer your question.
Yes, the Minnie Van driver meets you at the bottom of the escalator / baggage claim area just like a private car service. They will be standing there with your name on a sign with the other car service drivers.
Minnie Vans park in the same drive-through lane that is a floor or two below baggage claim that private car services use. It is extremely convenient.
FWIW, I found the whole experience extremely well-done by Disney. We've used car services in the past, some good, some not so good. The level of pre-arrival and day-of communication with the Minnie Van team blows away anything else we've experienced. I was actually quite shocked how well-coordinated the whole thing was... I expected it to be just a airport service slapped together, but they had put serious thought/attention into the customer experience details... from my perspective.
With the exception of the cost which is of course a major consideration, this service was a winner for us.
Happy to answer any questions.