First of all, it doesn't say anything about where I am. I may or may not be on the roof, but the bird is. So let's get that out of the way.
It's interesting (to me), because you've just made me realize something I had never thought about before. When I say "-daki", I immediately want to say "kuşu", i.e.; accusative. So, where does that lead us? Apparently, this -daki suffix makes the noun definite.
X'de (bir) Y → a Y on the X.
X'deki Y('yi*) → the Y on the X.
*if it's the direct object.
There might be exceptions to this, and "çatıdaki bir kuş" doesn't immediately strike me as an error, but I just can't come up with an example that also sounds natural. "Çatıdaki kuş" is fine, but with the "bir", it sounds weird. Furthermore, I don't think one could ever use it as the object. You definitely need an accusative if it's the object. That is for sure.
Let's wait and see what other natives think. (Deorme! We need help! :D)
Oh sure, you can express it in many ways. Yours is correct. Alternatively: Ben çatıdayken bir kuş görüyorum. (I see a bird while I'm on the roof). If you want to say that you only see the bird while on the roof and not elsewhere.
Or simply: Ben çatıdayım ve bir kuş görüyorum. (I'm on the roof and I see a bird.)
Thanks for the explanation. Very clear :)
However, I have a doubt : what about the construction X'deki Y when it's the subject of the sentence ? For example : The car in my garag is big. Would it be :
Garajımdakı araba büyük. -- or -- Garajımdakı arabayı büyük. ?
(And bonus, is the sentence "Rüyalarımdakı araba(yı?) var" is correct to mean "The car of my dreams exists." and I'm desperately seeking it all over Turkey ^^)
Thanks a lot ;)
Garajımdaki araba büyük.
Rüyalarımdaki araba var. is correct, but we would expand it a bit more to stress the var:
Rüyalarımdaki araba, (gerçekte) var olan bir araba. = the car in my dreams is a car that exists in reality.
Btw the -ki is not harmonized. It's always -ki, (except in dünkü and günkü).