One positive characteristic of these checks is that, if the depositor doesn't call the bank's attention to the fact that the check fits the definition (for example, a bank's treasurer's check and some forms of money orders are cashiers' checks by definition) by using a special deposit ticket or otherwise pointing out the check is a next-day item, the bank can handle it as a garden-variety check.
The definitions Brian provided are from the UCC. Regulation CC, however, tightens the definition a little. Under the UCC definition a bank's payroll check (if there are any banks out there that still issue them to pay staff members) would be a cashier's check. Under Regulation CC, it's not. Under the UCC a bank-issued bill-payment service check (if drawn on the bank itself) would be a cashier's check; under Regulation CC, it's not. The same goes for accounts-payable checks to vendors.
Check out the Regulation CC definition and official commentary on it in Reg CC section 229.2(i) and the definition and official commentary on teller's checks in section 229.2(gg) at
https://www.bankersonline.com/regulations/12-229-002.