What is "Expectation over $Y$" in the Law of Total Expectation

Question

I am struggling with the Law of Total Expectation, in the part of "expectation over $Y$".

In the lecture note, page 55, the Law of Total Expectation is stated as

$\mathbb E(X) = \mathbb E_Y\big(\mathbb E(X|Y)\big)$; where $\mathbb E_Y$ denotes expectation over $Y$.

My questions are:

1. What does the subscript $Y$ in $\mathbb E_Y$ mean?
2. Why in some other documents, they just simply write the law as $\mathbb E(X)=\mathbb E\big(\mathbb E(X|Y)\big)$ (without the subscript Y, for example in another lecture note, page 3)? So, does it really matter to keep the subscript $Y$ or just an option for a more clarified?

Any help will be appreciated. Thank you.