Shulchan Aruch 489:1 says that after Maariv of the second night of Pesach we begin to count Sefiras HaOmer. The Mishneh Brurah there in note 2 says that one should count before "Aleinu" because "col mah d'efshar l'akdomai makdinan", meaning that as early as one can possibly start counting, one should. He writes that this is in order to fulfill what it says "t'mimos tiheyenah", "they shall be complete": we want the counting to be complete and full, and therefore each day we count as early as we can, to have it be a "full day".
Why did the Mishneh Brurah use the language of, "col mah d'efshar l'akdomai makdinan" and not that of "zrizin makdimin", meaning one should be quick to do the mitzvah? The latter is normally the language used in order to get someone to do the mitzvah quicker, faster, earlier, etc. Is there is a difference in these two languages? Is there a reason the Mishneh Brurah chose one over the other?