I agree that "the umbrella" and "an umbrella" should both be accepted, but I disagree that "the" sounds more natural here than "an." As a native English speaker translating this sentence, I immediately pictured myself leaving the house and saying to my wife, "Wait! I forgot an umbrella!"
Hmm, but to your friend you would say "I forgot the umbrella!" since you have only one with you at a time (usually) and since you don't share any umbrellas with your friend like you would with your wife to pick "an" umbrella from :D
OK, I can make it to be equally probable for "an" and "the" to be displayed in this exercise.