My dad took a class in college where for one of their projects they had to keep a dream journal and wake up every 2 hours for that. So I think you are missing some key points:

You said the third time you got out of bed. Dont do this, just record in your dj whatever you remember from your dream. and I agree, maybe you should wait like 4-5 hours before having it go off. But again, the most important thing is to record your dream and fall asleep again. And no, it wont 'cut short' your dream, dream time and reality are two entirely different things.

Hope this helped