Edit: Oh, well, a much more localized answer was already given so never mind. I'll guess I'll not delete this, uh... just in case? In case of what, I don't know.
I'd write an invoice (everything being online), send it, receive the money by bank transfer, then send a similar document with "invoice" changed to "receipt" and a stamp saying PAID on it. First of all, is that right?
That much is essentially correct, although the language is usually '$X.XX RECEIVED' while 'receipt' is the name of the paper (or electronic document) that information is on.
Your receipts don't have to document everything (or, honestly, anything... the receipt does nothing directly
for you, it is given to your customer as proof that they are paid up and as such it is a service
to them. Providing good customer service is a good idea, of course, and indirectly builds goodwill... and if you aren't planning on some shady double-billing there's no reason not to give one.) Your
invoices on the other hand, are a paper trail of your demands in case there's any trouble with payment.
I haven't done business in Japan, but I don't see why it would be different... aside from being written in Japanese, of course, and being stamped instead of signed (which if it's all digital is probably actually just typing your name anyway in any language).