Just so everyone is clear, 'training' is defined as flight instruction which will lead to a new certificate or rating. Therefore and an initial Private Certificate requires registration with the TSA. As does a multi or instrument rating
Commercial training does NOT require registration unless you are obtaining a new certificate i.e. multi rating / instrument. Nor does a high performance endorsement, tail wheel etc.
For example if you already have a Private Multi Certificate and are training for a Commercial Multi Certificate you do not need to register with the TSA!
Any questions feel free to email me. I currently handle all TSA processing for a well known flight school in Florida and also work for a very well known GA publication in the UK!
Safe flying everyone.