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The Kedushas Levi answers that we should really say like Dovid as we have a rule from Nedarim that we do not say "for Hashem a Korbon" as you might die Mid-sentence and you have said Gods name for no purpose but here we know that after Kriyas Yam Suf Death was taken away until they sinned at the Egel there was no such concern and Hence the Change of language.
Again, context here gives the simplest answer.
At the Red Sea, it's just about G-d's mastery of the world. "G-d shall reign for ever and ever." Done.
Psalms adds the element of Jerusalem. (In fact, that's why this verse makes it into Kedusha -- the editors of the siddur felt kedusha would be remiss without some mention of Jerusalem.) So right after "G-d" is the explanatory phrase "your L-rd, Zion." Thus:
Reversing the order there would make for a more convoluted sentence structure. "Hashem shall reign -- your L-rd, Jerusalem ..."