RE: I can update those deposits to change the Received From to apply them to the customer, but that doesn't seem to apply them to the open, unpaid invoice.
That's correct. It won't. Based on this description, you didn't receive and apply Payment transactions as previously asserted, but instead recorded deposits. Notably, if you recorded deposits and used income accounts, and then recorded an invoice and also used income accounts, then you've doubled your income.
If you want to record invoices for your sales, then the proper sequence is Invoice -> Payment -> Deposit where only the invoice uses income accounts, and both the payment and the deposit will only use balance sheet accounts. Trying to do it backwards is somewhat difficult, but in this case you can record payments and then add them do your deposits, replacing the detail that is there.