It depends on the project - some charge commission on all purchases, others an hourly rate, or both. I'd be surprised if it's less than $50/hr for someone who isn't a big name, and over $100/hr if they are.
And yes, it's true that "designer" often means decorator and the terms shouldn't be interchangeable. A designer can sign off on certain aspects of plans for new spaces and renovations and has to know all the building code requirements, and a decorator is mostly hired for their taste and sources fabric/furniture.