What are the respective ownership rights between a website designer and a customer who has paid for the development of the site?

A creator of a work is the presumed author and owner of the work, and copyright law says that a copyright transfer has to be in writing to be enforceable, to protect authors.

You want to enforce an oral agreement for the development of a website as a specially commissioned audio-visual "work for hire" so that you own all proceeds of the website work. Even if your payment check or their invoice indicates that you paid for a "website design," it's unlikely that you could successfully argue that you have a written transfer of all rights to the work. It sounds like this company knows its rights and is asserting that you only paid for the images that they put on your site when they delivered it -- not their drafts, underlying material that they originated, other variations on their work, etc. Under prevailing copyright law, they have the better argument.

Disclaimer: Please note that this answer does not constitute legal advice, and should not be relied on, since each state has different laws, each situation is fact specific, and it is impossible to evaluate a legal problem without a comprehensive consultation and review of all the facts and documents at issue. This answer does not create an attorney-client relationship.

I agree that under copyright law the design company is right in its assertion that it "owns" the designs. What that means is that it owns the copyright in the designs.

But let's be clear: oral agreements to use a copyrighted work are most certainly valid. The issue in your case is whether the license the design company granted to you to use the designs on your website when it was hosted by web host one also permits you to use the licensed designs for the same website when it's hosted by web host two. I think it very probably does -- unless the design company is, or is associated with, web host one.

So while you don't "own" the designs you do own a license to use the designs. You need to hire an intellectual property attorney to review the situation and write a letter to the design company. And never again hire someone to do something important w/o having a written agreement in place. Good luck.