Blender's stabilization simply transforms the image by the track's neative position value and the Location Influence value is a factor that influences the shift.
This is not usable if you try to stabilize a video that has also intended camera movement like a pan, because that movement will also be compensated.
A simple refinement of the stabilization feature is to duplicate the tracker's f-curve, smooth the duplicate (which works as a lowpass filter) and then subtract the smoothed curve from the original. The resulting curve will only present the high frequency movement, but no slow pans and other intended movement so it can be used for stabilization.

Ways to ímplent this are either adding such an option with a user defined lowpass amount to the stabilizing feature or adding f-curve modifiers functionality to the tracker's f-curves so that the user can duplicate a track and smooth it's motion manually to subtract the curves with the compositor and so do a manual stabilization.