The best information I have on TNT Distribution Services, which is admittedly pretty weak, is that they were nominally associated with TNT Contract Services. They provided a small degree of warehousing services, via borrowing crossdock space from the TNT LTL's, but mostly acted as a product distribution provider. What many LTL's refer to as "trade show service".
The reason nobody knows who the hell they were is because the TNT Freightways carriers weren't really involved in it, and it was never widespread. As far as I can tell they were eventually quietly folded into TNT Contract Services, which rebranded as Logix/USF Logistics after TNT sold out. Then again as YRC Logistics after USF was purchased by YRC. Most (all?) of YRC Logistics has been parcelled out and sold off, including the portion that used to be TNT Contract Services.