Consider an orthogonal axis system (X,Y and Z). When the term "uniaxial" is used, it means "along one axis". This is generally taken to mean the long axis or loading in the "Y" direction. For soil with reasonable shear strength and for concrete, this loading is done vertically without "confining" or loading the other two axes, X and Z. Compare this to a triaxial shear test on soil where the X and Z axes are loaded to confine or constrain the sample with, generally, a constant load as the vertical load is applied.
fattdad...I wrestled with that while I was giving my long-winded explanation. On the surface, uniaxial is just one direction loading. If there is confining pressure, it becomes triaxial, even if the confining pressure is minimal.
. . . Ron, I guess you are right. I also figure I just use the term, "unconfined compression test." As the UU (unconsolidated, undrained TXC) is the means to add confinement, it'd clearly be a TXC by the ASTM.