I've been searching through this forum and have seen a few instances where 1005.3(c)(5) is cited that loan payment transfers from a deposit account within the same institution are not EFTs subject to the regulation. The way I'm reading Reg E and because a loan is not a consumer asset account, if a consumer chose to set up automatic payment from their checking account to their loan (at the same institution), a written authorization needs to be obtained and copy given to the consumer. In the commentary under 1005.(c), "loan payment" is not an example and doesn't seem to relate to the nature of the examples. Below are my cites (bold/italics are mine) and any help is greatly appreciated.

1005.10(b) says "Preauthorized electronic fund transfer from a consumer's account may be authorized only in writing signed or similary authenticated by the consumer. The person that obtains the authorization shall provide a copy to the consumer."

1005.3(c) says "Exclusions from coverage. The term "electronic fund transfer" does not include:
(5) Automatic transfers by account-holding institution. Any transfer of funds under an agreement between a consumer and a financial institution which provides that the institution will initiate individual transfers without a specific request from the consumer:
(i) Between the consumer's accounts within the financial."

1005.2(b)(1) says "'Account'” means a demand deposit (checking), savings, or other consumer asset account (other than an occasional or incidental credit balance in a credit plan) held directly or indirectly by a financial institution and established primarily for personal, family, or household purposes."

You missed one exclusion.

1005.3(c)(iii) Between a consumer's account and an account of the financial institution, except that these transfers remain subject to § 1005.10(e) regarding compulsory use and sections 916 and 917 of the Act regarding civil and criminal liability.

Thank you Randy!