If the transfers are covered by Reg E because they are performed through internet banking, then 1005.10(a) says you must provide a phone number for the customer to call to verify that the transfer occurred.
Internal transfers are not subject to Reg E so any notice requirements would be based on your deposit agreement. There is no federal regulation that requires notice each time a transfer is made.
Section 1005.10 only addresses pre-authorized EFTs (authorized to occur repeatedly), so the phone number info access doesn't apply here.
"Home banking equipment" is not an electronic terminal because it's analogous to using a telephone. Remember that much of Regulation E was written before home computers and the Internet became available. No receipt is required. See comment 2(h)-1.
A transfer between in-house accounts made in today's world using the bank's online banking portal is subject to Reg E and you need to include appropriate descriptions for those transfers on statements, but there is no receipt requirement.