You could've remembered the umbrella when you were in the driveway, getting into your car- so forgetting the umbrella "happened" while you were not really at home.
At any time it could have been remembered, but it wasn't- the act of forgetting is strewn all over the timeline, and basically occurred whenever the umbrella could've been remembered, but wasn't. It's more the case that remembering the umbrella did not occur. At no particular moment did the forgetting occur. You're getting ready, gathering stuff you're supposed to bring- the umbrella gets lost in the shuffle. "I forgot it" (in the simple past) sounds deceptively a completed action but it is not, it's an ongoing state- it does not have its beginning at a certain point in time. Besides, the definitions cited do not support that crucial additional detail of where the forgotten item was left behind. But this is a matter of idiom and definitional nuance, not the sort of stuff a book provides real authority on. I notice that almost all the people who think "I forgot it at home" is fine are not native speakers.
Explanation (for those who still need one):
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