worlds collide
It's funny because I do commissioned art and illustrations, and follow a forum or two dealing with art commission problems. I find I often cite how business is usually done in the reptile community since, in the artist forums, things tend to fall apart on shipping disputes, which I would say the reptile business community has down pat because it's such a big expense, risk, and important part of the transaction.
And now I'm going to cite what I know of how art commissions are usually done in a reptile community. XD I'm no lawyer but I've read up a lot on these things over the years.
I know both parties feel the problem is relatively wrapped up, if not entirely satisfactorily, but for the sake of anyone else considering these ventures, here's my two cents:
The problem seems to stem from lack of communication, contracts, and knowledge about legal matters on both sides. It is true that copyright is considered to automatically belong with the artist at the moment of creation, and that, unless otherwise specified, the person commissioning art only has rights to the original copy and has no right to reproduce or redistribute the art. It's also true that it can be pretty hard to defend copyright if it is not officially registered, and there is usually no money to be had in a suit unless there are demonstrable damages (such as items sold).
However, this is why logos are usually fairly expensive - because it is standard to include copyright. If a business buys a logo, they usually want to be able to use it on all their goods, and for a small business especially it would be very hard to keep track of royalties.
That said, licensing art for one type of item only is not unheard of. But it would be easy for a commissioner to get confused if it was not made clear ahead of time that the logo could only be used on business cards and nowhere else.
I could go on and on, but what it's going to boil down to is: OP, you were wrong not to pay for the website, and should have had the money socked away since you knew about what they'd be asking for it. It also sounds like you could have been better with communications.
Artists: always ask for some payment up front to avoid getting burned. Also to be proper professionals you should have terms of service (just like a reptile business!) and a contract for commissioners so that everyone knows exactly what they're getting, and if there's a dispute you have paperwork to back your side.
I would honestly say the greater burden is on the artists to make legalities clear since you are the professionals in this area.
From what I see here both sides messed up about equally, probably because both were in somewhat unfamiliar territory and trying to deal with "friends", and hence contracts and TOS were neglected.
Good luck.