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I like to say that someone “fell asleep” when they failed to perform at some kind of task due to inaction, often by lack of attention, because of distration, you forgot or, less often, failure to understand the task in the first place. If someone is “asleep”, they're not paying attention or show behaviors that look like they're asleep. In English you could say “I was asleep” to mean roughly the same thing.
In German we have the word “verschlafen”, which is a bit like “oversleep”, ie. you erroneously sleep past a specified moment during which it is your turn to do something. Though, in my context, sleep is obviously a metaphor for just being inattentive, distraction or forgetting.
Care should be taken when feeling offended by me claiming that you or someone is asleep or “overslept something” (I don't usually say this in English because it doesn't translate well), as this is mostly value judgement-less. I mean, sometimes I can be annoyed at someone being distracted, so me saying this as an expression of frustration is definitely realistic. BUT, more often than not I will actually use this to describe myself. I recognize that a lot of mistakes I make in day-to-day life are due to lack of attention or distraction, and I will say it exactly like that about myself as well. “Sorry, I was sleeping”, something like this.