Yes, it's agreed that they convey those meanings, so far as I know. The order of the quantifiers change the scope, and so change how we interpret what's being said. To explain: "*∀x∃y xSy*" - The existential quantifier is in the scope of the universal quantifier, so this is translated as "For every x (every student), there is some y (some person which exists) such that each student saw some person". In other words, for each student, it is the case that they saw a person. Think of the "There is some Y" as extending FROM every student. The existential quantifier is bound to the universal quantifier here. "*∃y∀x xSy* - The universal quantifier is now within the scope of the existential quantifier, so this is translated as "There is some y (a singular person) such that every x saw that person y". In other words, think of the universal quantifier as extending FROM the existential quantifier, from a particular person, to being seen by every student. I hope that was helpful!

Sure but Godel's work didn't, so far as I know, decimate the portions of Russell's work that Priest mentioned in the video. So its a bit besides the point, isn't it?

How long do you want the lecture to be? 3 hours? To go into immense detail in the format of a lecture wouldn't be practical. Go back and read the mathematicians/philosophers work in question to gain the best understanding possible of the ideas hinted at in the talk.