@Elspeth15
I am not 100% sure what happened here. Are you saying that someone made a third party booking, i.e. a user booked and then it turned out that someone else, not a user on Airbnb, was the person actually coming to stay? It's the only explanation I can think of, as I can't see how else someone who is not registered on Airbnb could book your place in the first instance.
Third party bookings are technically not allowed on Airbnb and will not be covered by Airbnb insurance, nor will they help you out if anything else goes wrong, so you were right to not want to accept the booking. However, I have been told by several Airbnb CS reps that it's up to the host if they want to take those bookings (and the risk). There is nothing to stop you accepting them.
In future, I would absolutely not cancel a booking yourself if you can help it. You should have first told the booking 'guest' that third party bookings are strictly against Airbnb policy and that they need to cancel from their end. If they refused to do so or didn't respond, then you should have contacted Airbnb before cancelling, telling them that you discovered it was a third party booking and asking for a penalty free cancellation.
This has happened to me a few times and Airbnb had cancelled the reservation with no fines or penalties to me. One time I got a rep who wouldn't do this, but she was a bit clueless, so I just told her I was going to hang up, redialled and got a more knowledgable rep on the line.
NEVER cancel yourself if you can help it. Either try to get the guest or Airbnb to do it for you, penalty free.
