Would any operation under general anesthesia be considered a major operation?

Not always: There are certainly operations that require general anesthesia that aren't considered major surgery. Examples would be quick laparoscopic procedures such as tubal ligation, or ENT procedures such as tonsillectomy. Also, GA may be needed for babies and small children who can't cooperate, even for small operations. The physician anesthesiologist works with the surgeon to make the best plan.