"Vort" refers to vorticity maximum. Vorticity is a measurement of spin in the atmosphere, where positive vorticity refers to counter-clockwise rotation, which in the northern hemisphere, is associated with cyclonic storms. In strict mathematical terms, it is the curl of the velocity vector. Associated with that, an important quanity is "positive vorticity advection (PVA)". Areas experiencing PVA are having the vorticity increase overhead, and that is associated with upward vertical motion and can encourage cyclogenesis. The "X" on a 500mb vorticity chart usually is the location of maximum vorticity within a trough.
"Shortwave" (sometimes abbreviated "s/w") is short for "shortwave trough", which I think is also your reference to "trough". However, shortwave troughs are distinct from LONGwave troughs. Shortwaves are embedded in a broader longwave pattern. When people say "we want a trough in the east for cold weather", they are referring to a longwave trough. Shortwaves are associated with positive vorticity maxima (in the northern hemisphere), so many times those 3 words you referenced are used interchangeably even if they technically mean different things. Shortwaves are often associated with baroclinicity (a potentially unstable difference in temperature) and thus can be associated with cyclogenesis and/or thunderstorm development.