How will I know that the hotel has actually received my customer's booking?

Yes, a hotel booking API does tell you when the hotel has the booking, and it does that by sending back a confirmation reference as soon as the reservation goes through. For most rates this happens within seconds of your customer paying, and the reply carries a supplier reference and, in many cases, the hotel's own reference too. Your portal stores both of those and prints them on the voucher your customer receives, so there is something concrete to quote at the front desk.

Not every rate is instant, though. Some hotels and some suppliers work on request, which means the booking sits as pending until the hotel itself accepts it, and that can take a few hours or occasionally longer. In that situation the API gives you a pending status first and the confirmation afterwards. Your portal should show that state honestly to the customer rather than telling them they are booked when the hotel has not yet said yes.

It also helps to know that the supplier reference and the hotel's own reference are not always the same thing, and whether you receive the hotel reference at all depends on the supplier and on how that particular property reports back. Where it is not passed on, the voucher together with the supplier reference is normally enough for the hotel to find the guest on arrival.

If you are not sure how your own bookings behave, make a test booking on a rate you can cancel free of charge, then look carefully at what the confirmation screen and the voucher actually show you, and ask your supplier which of their rates are on request rather than instant.