Why is Havdalah squeezed into another blessing instead of getting its own?
On Saturday night the Amidah has no blessing of its own for Havdalah. The Mishnah (the earliest written record of the Oral Law) records three opinions. The first puts it inside this blessing, the one about understanding. Rabbi Akiva says to say it as a fourth blessing by itself. Rabbi Eliezer says to put it in the thanksgiving blessing.
The Gemara (the rabbis' discussions built on the Mishnah) brings up Rabbi Akiva's way in Berakhot 29a, while discussing a shortened Amidah called Havineinu. It replaces the middle blessings, this one included. The Shulchan Aruch (the main code of Jewish law) says it is for a hard moment, like being on the road or worried about being interrupted. Shmuel said you may use it all year except on Saturday night and after a holiday, because Havdalah must be said in the understanding blessing.
A sage named Rabba bar Shmuel objects: then say Havdalah as a blessing of its own, as Rabbi Akiva says, and then say the short prayer. The Gemara answers with a number. We do not follow Rabbi Akiva in general, because the sages set eighteen blessings, not nineteen. In the short prayer, they set seven blessings, not eight.
Mar Zutra, another sage, objects too: why not fold "who separates holy from ordinary" into the short prayer itself? The Gemara gives no reply and says the question "remains difficult." The Shulchan Aruch rules like Shmuel: no Havineinu on Saturday night.
None of this says why Havdalah sits in the understanding blessing in particular. The Atah Chonen prayer covers that. What this passage shows is how firmly the placement is held. It is not its own blessing and it is not skipped in a shortcut. One objection is answered with a number, and one is left standing.
Mishnah Berakhot 5:2,Berakhot 29a,Shulchan Aruch, Orach Chaim 110:1Why does a prayer for understanding say "between Israel and the peoples"?
This line is one of the separations Abudarham ties to a verse, and this one comes from Vayikra 20:26: "I have separated you from the peoples to be Mine." He adds Shemot 31:13, where God calls Shabbat a sign between Himself and Israel. So on the night Shabbat ends, the difference is named.
What does the separation mean? Rashi (the main commentator on the Torah, France, died 1105) reads the verse as a condition: if you are separated from them, you are Mine. If not, you belong to Nebuchadnezzar (the Babylonian king who destroyed the Temple) and his friends. Rashi then brings a teaching of Rabbi Elazar ben Azaryah from the Sifra (an early collection of teachings on Vayikra). A person should not say "pork disgusts me." He should say "I would, but what can I do, my Father in heaven has forbidden it." The separation should be for God: you keep away from wrongdoing and accept the rule of heaven.
The prayer also does not begin there. The Bach says the first sentence, "You graciously give a person knowledge," covers every human being, and the nations share in that. Only then does it say "You graciously gave us knowledge of Your Torah."
A verse in the prophet Amos looks at being singled out from another side. It is not about this prayer. God tells Israel: "You alone have I known of all the families of the earth. Therefore I will hold you to account for all your sins."
In these sources, the separation is tied to Torah, to Shabbat, and to how a person lives. Whether that settles the discomfort this line gives some people is a fair question, and the sources here do not decide it for you.
Vayikra 20:26,Rashi on Vayikra 20:26,Abudarham, Conclusion of Shabbat,Bach on Tur, Orach Chaim 294,Amos 3:2How can I ask for a week with no sin when I know I will mess up?
Look at the Hebrew word: chasukhim. It means held back, and it is passive. The prayer does not say "I will not sin." It asks to be held back. Abudarham, explaining these words in a prayer that opens Havdalah over the cup, ties the word to Bereishit 20:6, where God tells Avimelech, "I also held you back from sinning against Me."
Why ask for help instead of promising to try? The Gemara (Kiddushin 30b) says a person's urge to do wrong renews itself every day. Rabbi Shimon ben Levi adds that it grows stronger every day and seeks to kill him, "and if the Holy One did not help him, he could not beat it." That is the Gemara's strong language.
So asking at the start makes sense. Sefer Charedim (a book on the commandments by Rabbi Elazar Azikri, a kabbalist of Tzfat who died in 1600) says every day brings a new test, so you ask for help at the start of the day. He then lists the start of the week, pointing to these words as they are said in Havdalah, and also the start of the month and the year.
Abudarham noticed a catch. Where our text has "clinging to awe of You," his has "clinging to the study of Torah." He imagines someone saying, fine, then I will go to sleep all week and not sin. That, he says, is why the prayer goes on to say clinging to Torah. Staying out of trouble should not mean doing nothing.
So the wording asks for a hand that holds you back, not a perfect record.
Abudarham, Conclusion of Shabbat,Bereishit 20:6,Kiddushin 30b,Sefer Charedim, The Commandment of Repentance 4:2