The translation with implied punctuation would be:
"And don't touch it: To eat it, like, "and in regard to their carcasses, don't touch... (VaYikra 11:8)", [The expression don't touch it is intended] according to the plain meaning of the text (נגע to touch or strike). And the true [meaning is], all who add [to the commandments], diminish (יגע to weary or to cause trouble to) [the whole Torah]; and this [addition of not touching found here] is not a fence for the Torah since she said that the Holy One, blessed be He said, "don't touch it."
This is referring to the prohibition of being a Ba'al Toseph, one who adds to the mitzvot of the Torah which is found in Devarim 4:2. The Torah as given by Moshe is perfect and is G-d's will. If one adds anything to it, it is no longer perfect and no longer G-d's will, meaning it is diminished.
Chava did not add anything to the Torah. She only repeated what she was taught by her husband, Adam. The commentaries and Midrash emphasize that this extra prohibition of not touching the Tree of Knowledge of Good and Evil was instituted by Adam. Because Chava was not present as a separate individual when G-d gave the commandment.
Only Adam was present (See Bereshit 2:16-17 and 2:20-24). She had yet to be built from Adam's body and soul. According to the Torah teaching about reincarnation, when a soul is incarnated into a new body, that new body prevents the soul from having knowledge of the previous incarnation.
The building and formation of Chava was a parallel of the diminishment of the Moon, which was originally the same as the Sun in size and luminosity. (See Bereshit 1:3 which states in the beginning of creation, on the first day, there was a single, unified light [אור], like Adam HaRishon. In Bereshit 1:14-15 it says that on the the fourth day the two great luminaries [מארת, מאורות] were created, like the building of Chava from the original unified Adam HaRishon. The defective spelling suggests that this divided structure of Adam and Chava was less than the original, unified form. But the Mem prefix [מ-אור] also emphasizes that they were from the single unified light. It also indicates that because it can be understood as a verb, that Adam and Chava have the potential to be sources of light to the world. And then, in Bereshit 1:16, the second light was diminished and made into a 'Kli', a vessel that would receive its light from the greater light of the sun. And that is what was done to Chava in Bereshit 3:16.)
And after G-d built Chava from the original, first man, Adam HaRishon, G-d told Adam that he was to teach her, like the Moon receives its light from the Sun. And Adam did not think Chava could be trusted to keep the simple prohibition of not eating, so he taught her not to eat from it, nor even to touch it. She believed that this was what G-d had commanded because Adam told her this. And that error on Adams part, gave the opening to the Nachash to cause Chava to transgress because the Nachash was also present before Chava had been built and formed from Adam HaRishon. It knew what G-d had actually commanded and challenged her on what she said was the prohibition about touching. And when she touched the tree and didn't die, it created a doubt in her mind about the prohibition in general. Her doubt, which was the consequence of Adam adding to G-d's prohibition, precipitated and allowed the sin to occur.
This is what the Chizkuni is discussing, as are all the other classical commentaries you mention.
Concerning your last question about the relationship between the citation in Bereshit and that from VaYikra, it is only about the word usage about touching (נגע). There is no additional meaning. The citation from VaYikra is discussing two ideas. The prohibition against eating (לא תאכלו) those non-kosher animals and the second prohibition about not touching (לא תגעו) the carcass of the non-kosher animal and becoming Tameh.
Like the commentaries mention there, the warning about not becoming Tameh was intended only toward the Kohanim (See Shulchan Aruch, Yoreh De'ah 369 and 373. Although some suggest that it may be applicable to all Jews at specific times (the three Festivals), as contrasted with the Kohanim, because the Yisraelim are supposed to go to the Temple at the time of the three Festivals; the constant warning against becoming Tameh (The prohibition mentioned in VaYikra 11:8 is undifferentiated as to when it applies. That means on the simplest level that it means continuously, just like the prohibition against eating non-kosher animals is understood to be continuous.) only applies to Kohanim because they were involved with Temple service constantly.
This concept about Tumah as it applies to Yisraelim (particularly in the present) is discussed by the Lubavitcher Rebbe in volume 3 of his Igrot Kodesh, page 374-375 beginning at paragraph 3.
And that detail, that this warning in VaYikra 11:8 about Tumah is intended for the Kohanim, is found in the Oral portion of the Torah which was also given by Moshe Rabbeinu. It is not an addition to the Torah.