Okay, I understand you now. I got confused because you said "measuring the angle", when in my case the angle is already known (and therefore doesn't need to be measured).
As for Wikipedia's definition, now that I read it again it actually seems to be correct, in a roundabout way. The article uses vertical lines as an example, but assumes the shear is along the X axis. In the case of perfectly vertical lines, the slope 1/m is either undefined or infinity, depending on how you want to interpret the division by zero. If they're measuring the angle of the oblique against the Y axis--in which case perfectly vertical lines are parallel, i.e. so that the slope is 1/Infinity = 0 (for our purposes anyway)--that might actually be correct. I say might because it's still very confusingly written, in any case--this is why Wikipedia is useless for any kind of mathematics research.
Trigonometry is fascinating!
edit + tl;dr: Wikipedia is measuring the oblique against the Y axis, not X. 1/tan is correct in that case.