Understanding $\forall x$ as for each or for all I find very confusing these different first-order sentences. Can someone confirm on my elaboration or if there is a technique to think about?

$\forall x \exists y\ x < y$, with domain $\mathbb{N}$ excluding 0 {1,2,3,4..}

For each x there is a y, therefore statement is true. However, if I think of $\forall x$ as "for each" I come to the statement above, but if I think of "for all", it goes "for all x exist a y" and this makes me doubt if the statement above is correct, because it sounds like that we want a y for all the x.
I draw here all the possible combinations to make my point clear and I will now just keep going considering $\forall x$ as a "for each".

$\forall y \exists x\ x < y$

For each y there is a x, therefore it is false (e.g x < 1)

$\exists x \forall y\ x < y$

Exists one x for each y, therefore it is  false
 (e.g 1<1)

$\exists y \forall x\ x < y$

Exists a y for each x, therefore it is false (thanks Andres)

I have often felt this is a problem, both for the ambiguity the "for all"/"for each" that you identify and because at one stage I was in an intuitionistic phase and less confident about making definitive statements about actual infinities.
My suggestions would be


*

*Use "such that" after "exists": for example your final $\exists y \forall x\ x < y$ would be read as "there exists a $y$ such that for each $x$ it is true that $x \lt y$", which is plainly incorrect for natural numbers, while without it "there exists a $y$ for each $x$ it is true that $x \lt y$" is less clear as to the intended meaning.

*Prefer "for each" (or  André Nicolas's suggestion of "for any") rather than "for all", so that in your initial example $\forall x \exists y\ x < y$ it is more obvious that you are looking at individual values, so this reads as "for each $x$ there exists a $y$ such that it is true that $x \lt y$", which is plainly correct for natural numbers.    
