If we authenticated a customer at the branch teller window by having the customer swipe a debit card and enter a PIN, and then allowed the customer to withdraw $200 in cash, would the transaction be subject to Regulation E, including adhering to the error resolution procedures and periodic statement disclosures? In this transaction, there would be no additional authentication, e.g., review of identification, or a wet signature required.

Yes it would. Resolution may be as simple as reviewing footage and positively identifying your customer, but it is still covered. What happens when a card is lost with the PIN? What happens when Jim steals John's card? What happens when a grown child takes mom and dad's card without permission?

Yes - these are still covered.

Maybe I'm wrong here, but I don't see this being an EFT. You used the debit card to identify the transaction, which was a teller transaction, the transaction itself was not an EFT. The term Electronic Funds Transfer is defined:

b) Electronic fund transfer. (1) Definition. The term “electronic fund transfer” means any transfer of funds that is initiated through an electronic terminal, telephone, computer, or magnetic tape for the purpose of ordering, instructing, or authorizing a financial institution to debit or credit a consumer's account. The term includes, but is not limited to:

(i) Point-of-sale transfers;
(ii) Automated teller machine transfers;
(iii) Direct deposits or withdrawals of funds;
(iv) Transfers initiated by telephone; and
(v) Transfers resulting from debit card transactions, whether or not initiated through an electronic terminal.

I don't see using a debit card as ID for a branch employee transaction would be covered.

Compare that to (v) - Transfers resulting from debit card transactions, whether or not initiated through an electronic terminal.

How about describing for us how the customer withdrew the $200. Was a separate withdrawal transaction made (with the swipe and PIN merely verifying identification)? Or did the customer, after the swipe and PIN, key in a $200 withdrawal request? Did the teller key in the request?