I don't think we need "the" at the beginning of the sentence. People sit down... is just fine.
It does make a difference in English and I assume in Hungarian also. It depends on whether you are talking about a specific group of people or any group of people. It's definite article vs indefinite article, except that in English, the indefinite article is omitted in a plural.
Can this verb refer both to the act of sitting down, and the state of being seated?
No, only the first. For the state of being seated you would use it without a preverb: "Az emberek ülnek az ebédlőben."
Without the "the" it is a general statement in English. With the "the" it is a specific group of people. You cannot tell which it is from the Hungarian so both should be accepted.