I do not remember when I first learnt of the word "Noahide," or where I first read that the Seven Commandments were a rabbinic distillation of part of Genesis. I probably asked Rabbi Google. Roughly ten years ago I found the relevant Torah passage in my copy of Dr J.H. Hertz's 1937 humash. Here is the conclusion of Rabbi Hertz's commentary:

Genesis Chapter IX, 1-17. The Covenant with Noah. The Seven Commandments of Man.

Rabbinic interpretation of these verses deduced seven fundamental laws from them: viz. (1) the establishment of courts of justice; (2) the prohibition of blasphemy; (3) of idolatry; (4) of incest; (5) of bloodshed; (6) of robbery; (7) of eating flesh cut from a living animal. The Rabbis called these seven laws "the Seven Commandments given to the descendants of Noah." These constitute what we might call Natural Religion, as they are vital to the existence of human society. Whereas an Israelite was to carry out all the precepts of the Torah, obedience to these Seven Commandments alone was in ancient times required of non-Jews living among Israelites, or attaching themselves to the Jewish community. [For verse 17 Rabbi Hertz epands the commentary: "This concluding verse of the paragraph stresses the idea that the Covenant was not only with Noah but with 'all flesh that is upon the earth'.]