I would like to create a separate outgoing e-mail template for closed status tickets, but noticed when I click Add Template, I don't have any Ticket Types available. Once you use a Ticket Type in another auto-reply setup, can you not use it again to create a different template based instead on either ticket states or priorities?
This shouldn't be the case unless you already used all ticket types for the Closed status for the specific auto-reply option you want to use. Try verifying if you already used the Closed status for this auto-reply option and which types you used for it. Also note that you might first have to select a ticket state before the usable ticket types are shown.