Sidurim (such as Rav Yaakov Emden's, but also modern nusach S'farad ones like ArtScroll's and T'filas Kol Pe) include a short prayer after various of the paragraphs of "Ezehu m'koman" asking God to consider the recitation of the paragraph as if the person praying had offered the sacrificial offering discussed in the paragraph. But the wording differs for different paragraphs: Why do we have "May it be your will, God, our and our fathers' god" for chatas and asham and just "May it be [your] will", with no mention of God, for ola, sh'lamim, and toda?
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I once heard - do not remember from whom - that we say Hashem's name at the Korbanos that are done for a personal Kapara such as a Chatas and Asham to invoke Hashem's mercy upon us.