Your math is off. It should always be the ACTUAL/TOTAL. In this case that would be the payment (after the fee)/cost of a single coin --> 961.63/12638.74 = 0.076085907BTC... Ergo, you're actually getting exactly what is expected (0.07608591BTC) from direct calculation using the correct numbers. There is no "secret fee", at least, not based on this evidence.

The site shows (on the right) that the coin amount is coming from the [total minus the fee]; whatever amount you enter is includes of the fee. In other words, you're only getting 961.63 worth of coin when you pay 1000 due to the fee, just as the website/receipt implies. If you wanted 1000 worth you'd need to pay something like 1046.