If you want to keep the customer, it's your fault. If you are ready to dump them, then it's their fault.
Realistically, if it is just a simple spelling error, and if there is only one error, it really shouldn't be a big deal to your customer. If you gave them an opportunity to proof read and they didn't catch the error, then most likely no one else would catch the error either.
I have had customer complain and then tell me that that they will use these because its no big deal and ask if we can correct the error for the repeat orders. Those are the customers that ya gotta luv.
I have had other customers complain about something even more rediculous, like "I just dont like the typestyle, after they approved it and it was printed. I explain that they approved it and that it is the same as their proof, and they get really snotty with me and demand that we reprint at our expense. Then they say "and by the way, since your are going to reprint it, can you also change bla bla bla". Thats the customer that you need to dump imeadiately. They are trying to abuse you.
I once had a customer to refuse to pay us because the paper had visible fibers in the paper itself. Shouldn't have surprised me any, the first time this customer ever came to my shop they pulled out a stack of basically identical business cards explaining which printer printed each one and why they were unhappy with each one.