You are having trouble with the meaning of "the name" among people in Biblical times especially among the Jews.
The “name” of Yahweh does not merely refer to the four consonants of God’s preferred name for himself, YHWH (called the Tetragrammaton). Rather, it refers to Yahweh himself. Old Testament writers at times refer to God himself as “the Name,” a practice still used by modern orthodox Jews. To avoid pronouncing the sacred consonants, these Jews substitute YHWH with the Hebrew ha-shem (“the Name”). Old Testament writers, in fact, use the same sort of substitution, at times even portraying “the Name” as a person or man (Isa 30:27–28; Ps 20:1, 7). That Yahweh can bring deliverance is a familiar idea, but four consonants do not protect the people of God—God himself protects them.
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Heiser, M. S. (2017). The Bible Unfiltered: Approaching Scripture on Its Own Terms (pp. 69–70). Lexham Press.
Thus, when Peter said "by the name of Jesus Christ of Nazareth," he meant you are saved by the person of Jesus Christ and Immanuel is the same person.
Note: you can get the same answer from the Greek and Hebrew lexicons. It's just not as easy to understand.
Also see:
There was and is a world-wide belief that the name of an object, man, or higher being is more than a mere label only incidentally associated with the one who bears it. The name is an indispensable part of the personality. One might say that a man is constituted of body, soul and name.
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Bietenhard, H. (1964–). ὄνομα, ὀνομάζω, ἐπονομάζω, ψευδώνυμος. In G. Kittel, G. W. Bromiley, & G. Friedrich (Eds.), Theological dictionary of the New Testament (electronic ed., Vol. 5, p. 243). Eerdmans
In Scripture reading in synagogue worship the tetragrammaton was read as Adonai, but in the schools הַשֵּׁם was used in quotations from Scripture. Sometimes this is also used for Elohim,
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Ibid. Vol. 5, p. 268
The fulness of the being and work of Jesus Christ may be seen in His name.
-- IbId. Vol. 5, p. 272