With respect to the "$\equiv$" symbol (my comment was getting long!):
This is more in the way of speculation, but it seems that "$a \equiv b$" is used in some (not all) contexts to denote "$a$ is identically $b$", or in other contexts to convey that $a$ and $b$ are essentially equivalent (with respect to some equivalence-relation, e.g. congruence modulo $n$, or geometric congruence, or truth-functionality, or... etc.), again, depending on the contexts in which it's being used.
This would be consistent with the frequent use of the symbol "$\equiv$" in introductory logic texts to convey that $a \equiv b$ holds (is true) whenever $a$ and $b$ evaluate to the same truth value. It's also consistent with what you've read: "$\equiv$" can be read as "can be replaced in a logical proof with...", in the sense that if "$p\equiv q$", then replacing every occurrence of $p$ with $q$ (or vice-versa) will not change the truth-value of any propositions thus impacted.
I know that I've used "$\equiv$" and "$\leftrightarrow$" interchangeably on this site, when answering, e.g., questions about propositional logic: partly for reasons related to trying to match the notation used in the question, and partly due to being careless and/or not necessarily knowing better.