The prohibition for Jews is usually derived from Deuteronomy 12:23:
רַ֣ק חֲזַ֗ק לְבִלְתִּי֙ אֲכֹ֣ל הַדָּ֔ם כִּ֥י הַדָּ֖ם ה֣וּא הַנָּ֑פֶשׁ וְלֹא־תֹאכַ֥ל הַנֶּ֖פֶשׁ עִם־הַבָּשָֽׂר׃
Only be stedfast in not eating the blood; for the blood is the life; and thou shalt not eat the life with the flesh.
You can see this derivation in the gemara in Ḥullin 101b-102a and in the halakhic midrash Sifrei Devarim Paragraph 76, as well as commentaries on the verse in Deuteronomy (eg. Rashi) and the codes (eg. Tur YD 62).
For non-Jews, the gemara in Sanhedrin 56b derives it from Genesis 2:16:
וַיְצַו֙ יְהֹוָ֣ה אֱלֹהִ֔ים עַל־הָֽאָדָ֖ם לֵאמֹ֑ר מִכֹּ֥ל עֵֽץ־הַגָּ֖ן אָכֹ֥ל תֹּאכֵֽל׃
And the LORD God commanded the man, saying: ‘Of every tree of the garden thou mayest freely eat.
The gemara says that this means that fruit is permitted, "but not a limb from a living animal (ולא אבר מן החי)". There is another opinion there (on 57a and 59a) that derives it from Genesis 9:4, as you state in your question. To explain this derivation, Ramban writes:
אך בשר בנפשו שהיא דמו לא תאכלו כי נפש כל בשר דמו הוא
But living flesh — [and by life we mean] its blood — you shall not eat. Because the life of all flesh is its blood [quote from Leviticus 17:14].
He explains that the blood is an aside expressing what its lifeforce is, but the main point is that if the flesh is living (ie. it was connected to a live animal), it cannot be eaten.
Obviously, this verse can be understood in other ways (as you've demonstrated in your question), but this is the way the gemara wants to take it.