If ten or more men ate together, the Zimun is recited with the Name of Hashem, and the leader announces, "Nebarech Elokenu She’achalnu Mi’shelo" ("Let us bless our G-d from whom we ate").
The Tosfot Yom Tob (Rav Yom Tob Lipman Heller, 1579-1654), in his commentary to the Mishna (Masechet Berachot, chapter 7), notes the distinction between this text and the text used when one receives an Aliya to the Sefer Torah. The person who receives the Aliya begins by announcing, "Barechu Et Hashem Ha’meborach" – "Blessed is Hashem, the blessed One." Whereas in the Zimun we refer to G-d with the Name "Elokim," in the context of an Aliya to the Torah, the Name of "Havaya" is used.
The Tosfot Yom Tob explained based on the famous Rabbinic tradition associating the Name of "Havaya" with G-d’s mercy and compassion, extending beyond what is strictly deserved, and the Name of "Elokim" with strict justice. In the context of Birkat Ha’mazon, we speak of G-d’s providing His creatures with sustenance. The Tosfot Yom Tob notes that in a sense, G-d’s providing us with food is something which is strictly required. After all, if He created us, it is only right that He sustains us. Just as a father who begets a child is required to feed and care for the child, G-d, the Creator of all beings, is required to feed and sustain them all. Therefore, as we introduce Birkat Ha’mazon, we use the Name "Elokim," which signifies strict justice.
The Torah, however, was given to us as an undeserved gift. It was an act of great kindness that G-d chose us as the "administrators" of His sacred Torah. This is something we did not strictly deserve. And therefore, when a person is called to the Torah, he uses the Name of "Havaya," the Name associated with divine mercy and kindness, through which we were given the great privilege of receiving the Torah.
