Why are Lot's descendants considered among the Children of Israel, while Ishmael's descendants are not, even though Ishmael is more closely related to Isaac?

Jacob, son of Isaac, son of Abraham, is Israel, and he had twelve sons, one of whom was Joseph. The lineage of all the Children of Israel traces back to one of Jacob's sons, which is why Allah addresses them as "Bani Israel" (Children of Israel). Al-Tabari mentioned that the meaning of Israel is "Abdullah" (Servant of Allah), while Ibn Kathir stated that it means "O son of the righteous, obedient servant of Allah." As for Ishmael, he is Jacob's uncle, and Lot is Abraham's nephew, and neither of them is from the Children of Israel, because the Children of Israel are exclusively the descendants of Jacob.