So I'm thinking according to the regulation, if a bank does not provide a special deposit slip for state or local government checks, and certified, cashier or teller checks, then the bank is not held to next day availability of those items when deposited in the overnight/ATM?

Any help will be appreciated

If the bank requires the use of a special slip, it has a duty to inform customers of the need for the slips and make the slips reasonably available. See Comment 229.10(c)-6.c

So, if they do not require the use of a special slip, the bank must still comply with the next day availability of the checks in the original post?

If the bank doesn't require the use of a special slip to obtain next-day availability, it must nonetheless provide next-day access on the items detailed in 229.10. And that means the bank will have to have a procedure for identifying cashier's checks, teller's checks, certified checks, and checks drawn by a state or local unit of general government, none of which are identifiable based on MICR-line information.

So, if the bank doesn't require the use of a special ticket, it is left with two alternatives: (1) inform depositors how to prepare their "standard" deposit tickets or envelopes so as to identify those items for which next-day availability is required; or (2) provide for individual eyeball scrutiny for each item in deposits to identify such items.

From what I'm reading, the next day availability would go into affect for nonproprietary ATMs the day after the deposit is picked up, not when deposited, is that correct? Thank you so much for your continued help!

For non-proprietary ATMs, look to 229.12(f).

For off-premises proprietary ATMs (the ones you've been reading about), the deposit is considered received when removed from the ATM (229.19(a)(4)), but only if deposits aren't removed for processing more than twice a week. If deposits are pulled daily for processing, you treat the off-premise machine just as if it's in your branch.