As I was taking apart the trem, I instantly understood what was going on, when removing the first saddle.
The saddle is fixed into place via the intonation screw, in its front, and fine tuner in the back, which pushes down a spring against from the top to the base of the saddle.
Normally, the "part with the hole", which is the base of the two-piece saddle, rests flat on the base plate.
However, when the intonation screw is tightened in the front, it pivots in the middle and raises the back of the saddle. I don't know if it is because of flexing or because the base is simply not flat enough.
So, the workaround would be to set the intonation screw in that rearmost hole, when possible, and tightening it just enough to prevent the saddle from moving forward under the tension of the strings. That shall mitigate the difference in string height.
Thank you all for the help and attention. I wish you guys a happy New year.