Do regulations require that transactions initiated through Internet banking contain identifying information on customer statements, such as that which is required for ATM and ACH originated transactions?

On customer statements? I'm not sure.

Are you asking because they are there or because they are not?

I am asking because they are not there. When Internet banking started, it was usually a matter of transfers between savings and checking, with one signer doing most of the banking. With increased usage and movement between multiple accounts, it is difficult to track what accounts are affected by transfers.

What you're looking for is §205.9(b) of Reg. E. What's required is the type of transfer and the type of account to or from which the transfer was made.

So, a line on the checking statement that said "TFR to SAV" would suffice for the regulation.