Is there a current Texas rule that ATM withdrawals can be made from available balances only?

I don't think so. I know some banks will allow ATM withdrawls to include the overdraft limit when using that banks ATM machine.

That will be set up in the card issuing bank's parameters, not at the ATM. If you (as the bank issuing the debit/ATM card) choose to use the ODP limit in the balance the ATM network checks, your card holders will be able to pull from their OD protection.