You mentioned this verse in passing, but as far as I can tell, it provides complete and convincing proof that G-d is known by multiple names.
Exodus 6:2-3, from Mechon Mamre:
וַיְדַבֵּר אֱלֹהִים, אֶל-מֹשֶׁה; וַיֹּאמֶר אֵלָיו, אֲנִי יְהוָה.
וָאֵרָא, אֶל-אַבְרָהָם אֶל-יִצְחָק וְאֶל-יַעֲקֹב--בְּאֵל שַׁדָּי; וּשְׁמִי יְהוָה, לֹא נוֹדַעְתִּי לָהֶם
Translation from Chabad.org:
2. God (Elo-him) spoke to Moses, and He said to him, "I am the Lord (YHVH)
3. and I appeared to Abraham, to Isaac, and to Jacob with [the name] Almighty God (E-l Sha-dai), but [with] My name YHWH, I did not become known to them."
This is a verse that effectively synonymizes three names: Elo-him, YHVH, and (E-l) Sha-dai.