Notations: use of parentheses with "mod" and the "|" symbol

The $|$ operator represents divisibility, not logical OR. $a \mid b$ is shorthand for $a$ divides $b$. As for your first question, $a$ mod $b$ represents the remainder when $a$ is divided by $b$ but $a_1 \equiv a_2 \pmod b$ the $\pmod b$ modifies the equivalence relation.

Just a linguistic addition: the statement “$a\equiv b\pmod m$” was originally read (and some of us still read it) as “$a$ is congruent to $b$ modulo $m$”. It’s Latin, and “modulo” means “with respect to the modulus”. You should think of the the “$\!\!\pmod m$” as an adverb telling you in what way $a$ is congruent to $b$.