Analysis question on quantifiers

Something really important that I have learned is that the order in which one writes the quantifiers is of paramount importance; writing them in the wrong order can affect the truth value.

Now here's my question: Am I looking at quantifiers in a correct way? Let me give an example:

This says that for all possible x values, there exists a y. For every x value, there is at least one y that satisfies whatever..

Then you have:

This says: some y value such that for every possible x... whatever. To me, this makes y appear more "fixed" than in the first statement, since in the first statement, the y is in the context of all x.

Finally, I have encountered statements like:

How must I look at such a statement?

I have posted a lot of questions about this topic, and I feel like I'm beating a dead horse. However, I really want to ensure that I understand this topic well. I've never taken a class like this before, it seems like a whole new way of thinking.

Thanks