European warmbloods have had very well controlled and managed registries, in some cases for hundreds of years. If you bree a registered, let's say, Trahkener, to another registered Trahkener, the offspring is a Trahkener, but the offspring cannot be used for breeding registered Trahkeners until it passes a very rigourous inspection and is accepted by the registry. This is why the European warmbloods are very consistent in appearance and performance. One of the interesting things about the European system is that they will accept a horse regardless of its breeding IF it can meet the rigorous inspection standand, and it's not uncommon to have a horse accepted into multiple registries.
A TB/Belgian or a TB/hanoverian are a draft cross and a TB cross. I used to call my draft cross a "luke warmblood" and the term American warmblood is tossed around, sometimes half-jokingly as well.
The American Warmblood Registry is an attempt to define a type and produce consistent results like the European registries, but it's in its infancy. It will take decades, if not longer, for it to produce as recognizable a type and consistent performance results as the Europeans, IMO. I just scanned their website, and they have a lot of European warmbloods list as approved sires.