Gibson ES 339 - Seems to go out of tune when applying normal pressure when playing a chord. If I use very light pressure, not normal, then it is in tune. Could this be a problem with my nut ?
It is unlikely to be a problem with your nut. If your string action at the nut is high it can affect the intonation of playing in open position, but since you describe lightening your playing pressure, it is more likely that you are deflecting the strings too much when you fret your notes.
Light gauge strings and new, high frets on an electric guitar can be difficult to not over press and bend the notes. This is especially true if you learned on an acoustic guitar, or a badly setup electric.
It is common for players to apply too much pressure when playing. I did for years, and was out of tune all the time until someone mentioned it to me. Now as a teacher, much of the technique work I do with students is getting them to play more relaxed, and grip the instrument less.
One of the technique practices I use is to have the student play to much too less pressure, not making the notes come out cleanly, then slowly practice adding just enough pressure to voice the notes against the frets.
You can check your nut action by checking that the distance between the first fret and the bottom of the strings isn't more than the thickness of a medium or thick pick.