My invoice says AED but my card was charged in my own currency. Which one is correct?

Both, and they are answering different questions. The price is agreed in dirhams and the invoice is a UAE tax document issued against a UAE tax number, so the document is in AED — restating a tax document in another currency would make the paper and the screen disagree. What actually leaves your account is your own currency, converted at that day's rate, and the invoice carries a line naming exactly what was taken and at what rate so you can tie the two together.