Volk shares its root with English folk. Not surprisingly, it has the same meaning as the English word (as defined at m-w.com) in several aspects.
In a plain and general sense, it divides people into groups of culture, history, descent or language.
The word, however, covers much more meaning than this. The population or citizens are a Volk (das deutsche Volk). And so is a herd of animals, a swarm of insects etc. (Bienenvolk).
And the working or lower classes (the common people) are called Volk (gewöhnliches Volk). In the latter case, you can interchange the word with the other words.
Except for the sense of "lower or working classes", there's a plural: "Völker".
The difference between Menschen and Leute is already well covered in the question you linked. And there's also a question covering the difference between Personen und Menschen: "Personen" or "Menschen".
Hence, here's just a very short summary:
Leute is basically a collective noun and means people in general. It shares some meaning with Volk and Menschen when it comes to the sense of "common people".
But it can be used for any group of random people. Like "guy" in "you guys" or "folk" in "Hey folks". It's also used for family or friends ("Meine Leute"). Both senses are unique to Leute and neither Volk nor Mensch(en) can be used here.
Finally, Menschen is the plural of Mensch which represents a human individuum. Both Volk and Leute cannot refer to a single person.