According to the Hadith, Salaat should be performed immediately the time begins (Awwal Waqt).

Answer

Yes, it is proper. Your understanding is incorrect. Follow the times stipulated by the Math-hab to which you adhere. According to the Shaafi' Math-hab, the Athaan and the Salaat should be in Awwal Waqt. As soon as the time for the Salaat enters, it should be performed. According to the Hanafi Math-hab to which we adhere, the Fajr Jamaat Salaat should be when it is somewhat light, not in Awwal Waqt. It is improper and deviant to extract a Hadith from a book and then submit it to personal opinion. Follow the Math-hab of which you are a follower, then you will be on safe and sure ground. You are not a Mujtahid.