In constructive and intuitionistic logic, it is common to take $\bot$ as a primitive proposition, and to abbreviate the formula $A \to \bot$ as $\lnot A$. That is, the negation of $A$ is a taken as an abbreviation for “$A$ implies absurdity.” In this formulation, notice that $\bot$-introduction is simply shorthand for $\to$-elimination (modus ponens), since
$A$, $\lnot A$ / $\bot$
is the same as
$A$, $A \to \bot$ / $\bot$
This also means that $\lnot$-introduction is the same as $\to$-introduction. If, after assuming $B$, you can derive $\bot$, then you can terminate the discharge the assumption $B$ and conclude $B\to\bot$, but by the definition, this is exactly $\lnot B$.
Although it's not common practice, as far as I'm aware, to define $\lnot A$ as $A \to \bot$ in classical logic, it can, in my opinion, make some of the naming conventions a bit clearer.
To get classical logic from intuitionistic logic, we add the axiom schema $\lnot\lnot A \to A$, and then reductio ad absurdum (RAA) is simply a derived inference rule:
Suppose $\lnot A$. Derive $\bot$. Then conclude, by $\to$-introduction, $\lnot A \to \bot$. This is, by definition $\lnot\lnot A$. By the axiom schema, $\lnot\lnot A \to A$. Then, by $\to$-elimination, conclude $A$.
Unfortunately, as you've noticed, there is lots of variation among authors as to what different rules are called, and you'll always have to be aware of what different authors mean when they use a particular terminology. For instance, this recent question links to a handout in which a professor defines some natural deduction inference rules. In it disjunctive syllogism is called $\lor$-elimination, both modus ponens and modus tollens are called $\to$-elimination, reductio is called $\lnot$-introduction, and there are three different things called $\lnot$-elimination!