This is according to the NACHA Publication, "Third Party Senders and the ACH Network", Third Party Senders are a subset of Third Party Service Providers (TPSP). In other words, a Third Party Sender is a always a TPSP that acts on behalf of an Originator, but a TPSP does always act as a Third Party Sender. A TPSP is considered to be a Third Party Sender ONLY when it acts as an intermediary between the ODFI and the Originator and there is NO contractual agreement between the Originator and the ODFI. When an Originator uses the services of a TPSP but has also executed a contractual agreement directly with the ODFI, the TPSP is NOT considered to be a Third Party Sender and is not subject to the rule provisions governing Third Party Senders.
In short, it all depeneds if there is an agreement in place between the ODFI and Originator.