I know about the 10k limit if you deposit cash where you are required to fill out a form. But what about a bank to bank wire transfer? Is there a limit to how much you can receive before you have to fill out a form? And how does this apply to a personal or business type of bank account?
There is no limit, and no form to fill.
Potentially, the bank reports the transfer, but as the person receiving the wire transfer, you have to do nothing special.
Note that there is a chance the IRS contacts you as the source of the money is not clear - if it is income, it is taxable, and if it is a gift, it is also taxable (for the giver); so you need to either have it on your next tax filing, or you have an explanation why you don‘t owe taxes on it (for example, it comes from your own account, and you already paid taxes on it).
For purposes of reporting, wire transfers and deposits of coin and currency are reportable, regardless of domestic or foreign. The bank maintains its "know your customer" obligations and still is obligated to report transactions. Banks are obligated to provide as much information as they have available. There are two levels or reporting. The first could be thought of as standard reporting and the second for suspicious activity. Anyone suspected of controlling over $10,000 in a foreign country must have all of their transactions reported.