you can initial start your training and then once your fingerprint have been approved,
Not true.
The email from the TSA states :
THIS EMAIL DOES NOT ALLOW YOU TO BEGIN FLIGHT TRAINING, NOR DOES IT INITIATE ANY WAIT PERIODS.
then when you get clearance it will say :
All required information, including fingerprints, associated with the following Request for Training has been received.
Based on information available at this time, the Transportation Security Administration grants Permission to Initiate Training for the following flight training request:
Then once they have the results of your background check it will say :
Based on information available at this time, the Transportation Security Administration grants Final Approval for the following flight training request:
In summary no flying until they have received your fingerprints - that doesn't mean upon completion that means once the TSA receives them - normally a day or two later.