This is mostly correct. It’s also the case that “dreams” are formed after you wake up. You aren’t dreaming while you are asleep, your brain is firing random shit that makes no sense. As soon as you start to wake it tries to piece together what the fuck was going on into something resembling a narrative. This piecing together is part of the waking up and not a part of the sleeping. This is why you can have a dream about an alarm going of for seemingly tens of minutes or even hours, while you are being woken up by your alarm going off. Your alarm probably hasn’t been going for more than a few seconds, but your brain incorporates it into the narrative. Now this isn’t to say you can’t have a bad dream or nightmare and be woken up by them. The random firing can definitely cause enough stress to wake you up. Especially if you are ill (fever dreams) or under a lot of stress in general, your brain can misbehave during sleep and wake you up. It’s just that the “story” part of the dream only happens when you wake up, while you are sleeping it is random.