And sometimes it doesn't work. Patients are then given the option of going ahead without sedation, or rescheduling for general anaesthesia.

Hmm, I got one and they used ketamine and an amnesiatic. Not considered general anesthesia but I don't remember a thing. This was in the US.

But even "general anesthesia" still has differing categories.

Surgery uses propofol plus a gas anesthesia for "general anesthesia", but it is considered "general anesthesia" even if only propofol is used.