Why does gamma correction of image use two different definitions of gamma?
The concept of the gamma (and its correction) has emerged as a means of controlling and correcting the nonlinear voltage-brightness characteristics of the legacy input cameras and legacy display CRTs which are both analog devices. 

Both of them had nonlinear relationships between the optical brightness levels being captured (or displayed) and the corresponding voltage levels that result of (or driving) it.

In order to reproduce the image brightness levels (and RGB channel intensity levels as well) as faithfull as possible to the original content, the exponential distortions introduced by the CRT displays are pre-corrected by the studio processing equipment before the content is being transmitted for broadcasting. The correction being applied at the studio is the inverse of the distortion being created at the CRT display, hence they will cancel each other, yielding a faithful, brightness neutralized, image at the CRT output. 

Furthermore, for adequately displaying video content recorded with different standards and/or captured with different gamma correction parameters, the TV sets and monitors are available with gamma selection options as well.

Hence you can apply a gamma correction either at the camera input stage before it's transmitted or gamma adjustment at the display (CRT) stage during it's played back. So the difference you percieve possibly refers to the stage at which it's meant to be applied. So as you have stated, the same gamma parameter value will create two opposing results on the same image according to which stage it was applied to. Or stated in other words, a given value $\gamma$ will create the same effect with $1/ \gamma$ when applied at opposite sides.