Please tell me if I have this correct: Customer calls the bank to ask the teller to debit his account to make his loan payment. The teller writes up the tickets and runs them through the system. This would not trigger a monthly statement?

Customer calls the bank to ask the teller to debit his account to make his loan payment. The teller does virtual tickets through the teller system. This would trigger a monthly statement?

I am not sure why you think it makes a difference?

So does that mean either way it counts for Reg E and triggers a statement?

How the teller processes the transfer is not relevant. In either case your customer called and spoke to a person so they did not initiate a transfer though electronic means.

1005.3(c) Exclusions from coverage. The term “electronic fund transfer” does not include:
(6) Telephone-initiated transfers. Any transfer of funds that:

(i) Is initiated by a telephone communication between a consumer and a financial institution making the transfer; and

(ii) Does not take place under a telephone bill-payment or other written plan in which periodic or recurring transfers are contemplated.

It is a transfer initiated from a phone call - how you accomplish it operationally does not really matter. Since you likely are not delivering monthly loan statements to the borrower, you would have to send the monthly statement on the savings account.