If A ∪ (B ∩ C) is equivalent to (A ∪ B) ∩ (A ∪ C), why are the possible element locations stated more explicitly in the second set than the first? If x ∈ (A ∩ C) then x ∈ A, which is an option given by A ∪ (B ∩ C), so saying that x ∈ (A ∩ C) logically holds for both sides of the equation. However, it feels significant that I can say x ∈ (A ∩ C) when looking at (A ∪ B) ∩ (A ∪ C), but not when looking at A ∪ (B ∩ C), yet these are supposed to be equivalent forms of the same logic. Shouldn't the exact same information be available from both versions if they are just different ways of writing the same thing? Or am I missing something?
I think this just comes down to the fact that some notations are simpler (for certain purposes) than others. For example, compare prime factorization and decimal representation of natural numbers. It's easier to tell which of $13124$ or $32581$ is bigger, whereas that could be difficult to determine using prime factorization; conversely, prime factorization makes it trivial to compute greatest common factors.

Incidentally, I don't find either notation much more explicit than the other; this sort of thing is pretty subjective. I'd say that the LHS makes it easier to tell when an element is in the set, while the RHS makes it easier to tell when an element is not in the set.
If x ∈ (A ∩ C) then x ∈ A


Right here is where you are missing something. Neither of these expressions guarantees $x \in A \cap C$. Not sure where you got that from.

As an example, let $A = \{1,2,3\}$, $B = \{3,4,5,6\}$, and $C = \{5,6,7\}$

Then $A \cap C$ is empty, whereas both sides of the equation evaluate to $\{1,2,3,5,6\}$.