So, I remember more dreams when I am woken up around 7 by noise in the house, then I fall asleep, and then I remember dreams. So, I have a question. Am I remembering my dreams because I’m falling asleep right into a REM period, or because I generally wake up around 9, and I’m waking up from a REM period?
Sounds like a WBTB. That’s normal. I think it’s because when you fall back to sleep for the 2 hours (or how ever long) you go into REM immediately, more often, and for longer then the rest of the night.
Or did you mean that you remember more dreams at 7? 
If that’s it, you probably time your sleep cycles so your in an REM period at 7.
I hope I helped. 
From what I understand REM doesn’t happen at the depth of sleep. You have a period of deep sleep that has no dreaming in it. You dream in the period either before or after (more often after) The deep sleep cycle. So if you are wakened suddenly, you are probably still in the cycle which is why you can remember the dream better. If you wake naturally, you have given yourself time to get out of the cycle. Therefore it takes your brain more energy to remember it.
I don’t think remembering dreams has anything to do with the moment you fall asleep. So it’s probably because you wake up at 9 from a REM period.
I think that it could be because you fall back to sleep after a short wakeup. I know that that happens to me. If I think about LD’ing durring the short wakeup, or just rembering dreams, then I tend to rember at least one dream when I wake up later and have to get up.
For me I think it is easyer for me to keep the idea to rember dreams if I am not going into a long deep sleep after thinking about it, just going into REM and then wakeing up again.
don