Hey, Newbie here, but I've read through this and have to comment. Rog/Badsnap mentioned changing the tension between the nut and bridge but not changing the pitch. I don't want to create an argument, but unless the sting was changed, if the speaking length is the same, that isn't the case. I don't know who Perry is, but Rhodes56 was correct and I liked that post.
For any stringed instrument, the speaking length of the string is what you are tuning. On an electric guitar, the transverse vibrations are the only ones the PUPs will amplify, and that speaking length is from nut to bridge (you hope).
However, the strings also have a longitudinal mode. Remember, their length changes as they vibrate. The speaking length of this longitudinal mode does not always terminate at the bridge and the nut. It will be transmitted to the guitar body, and will likely be more obvious when you are not amplified because when you only amplify transverse modes, you make the LM mode smaller in proportion. I suspect the magnitude of the LM that can go from string LM to body, and then create a TM that the PUPs sense, is very small.
If some of the overtones you don't like or feel are changed, when the bridge to nut scale is unchanged, it is quite possible they are LM's. Changing the guage of an unwound string will alter the LMs. For wound strings the LM will be related to core/winding ratios, so you may not need to change gauge, but simply the brand of strings. Jim Ellis of the PTG (piano technicians guild) wrote a nice book on LM's, and you may be able to find some stuff by searching for Longitudinal modes and strings. There are some articles I've read from MIT so add that too if you like.
Anyway, for TMs, changing the tension will change the pitch. It will have less of an effect on the LM IIRC, so one way to check this is to detune the string slightly and see if the overtones stay on pitch. Often LM's are out of tune with the TM, and if they are close to an odd partial, the detuned string might sound better. You leave the LM on pitch and move the TM partials away from it so they don't beat.
If the issue is/was a longitudinal mode, you have different options on trying to modify it.
Regards and good luck,
Todd