And all of this is when the noun עם (people) comes by itself, that then it also comes when referring to a collection of individuals. But when it says עם ה׳ and moreover עמך with a pronoun referring to God – it includes all of the people with their priests.

And moreover, sometimes the noun ישראל (Israel) comes versus the noun בני אהרן (sons of Aaron). But when it says עמך בית ישראל (Your people, the house of Israel) – it also includes priests.

And עמיו (His people) is plural for respect, like אדוניו (his master). And its plural is בעממיך (your peoples) with a repeating of the letter מ.