Wright was a great generalist. He incorporated many formal elements that would become associated with the Modern movement (such as widespread structural steel/ manufactured glazing units etc), but this was mainly due to the fact that these were relatively new technologies and, as any good designer does, were implemented. I think that this is where you're drawing your parallel..
But.. I would not for a second group Wright into the "Modern" movement. His fundamental philosophy differed largely, especially on the topics of environmental engagement/ornament/and his research's effect on the greater urban fabric (you didn't find him theorizing to steamroll half of paris for residential towers :D )
I would group the examples you gave into more of an arts+crafts category. The Larkin Building can refer back to my first paragraph.
The Bauhaus was definitely a hybrid movement that shared many of Wright's ideals, but he was never affiliated with them and even then Wright's influence in selective and subjective. That being said, I don't really consider the Bauhaus to be true Modernism... it's an early variant that propagated some of High Modernism's key players.
I believe that attempting to classify architecture/art phases are largely futile. It's a very subjective thing, and loosely defined. But no, I would not consider Wright to be of the same vein as Corb/ Mies or any of the more traditional "high modernists" during their careers at that period.
