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Normally, you'll need something more than a payment confirmation. That's because it doesn't state what you bought or did. An invoice isn't always necessary, but you do need something with details from what it was and how much it cost.

So it doesn't always have to include your vAT number?

If you are not going to reclaim vAT (which is not applicable, because it is from New York, so there is no vAT anyway), then the vAT number on the document is not important, if I understand good.

If you just want to claim the expense, you simply need a document with a description of what you bought and the amount you have to pay for it.

So I don't think a vAT number is necessary in this context. No, those guys don't even know what a vAT number is. That's purely European vAT; outside Europe, many companies don't know what vAT (or a vAT number) is.