Receipt and Invoice
If I am in Starbucks, bought a coffee, already paid and need a proof of payment, with my name or company name.
Do I need to ask them "invoice" or "receipt"?

Question

Answer

Ask for a receipt - it's to show that you have paid that amount.

An invoice shows the amount that is requested - a bill.

Answer

An invoice is a request for payment.
A receipt is the proof that you have paid.

However I don't think that most shops would provide you with a receipt with your name and company on it. The receipt is typically produced by a cash register which automatically prints the amounts.

An invoice would probably have those details. Invoice may involved data about your name and business which someone inputs into an accounting system.