Towers of Hanoi Algorithm "Using Auxiliary" Peg
In mathematics, the literal meaning of a phrase is of little importance. What is important is that both author and reader know what a phrase means, and this can come about in two ways. Either the phrase is well-known, or the phrase is defined in the article. In your case, the phrase "move $n$ pegs from $A$ to $B$ using $C$" means "call the function Hanoi with parameters $n,A,B,C$" - no more, no less. The author could have chosen a completely different phrase, for example "jump from $A$ to $B$ doing $n$ flips above the midpoint $C$", as long as they make it clear that it means "call the function Hanoi with parameters $n,A,B,C$".

Another point that you raise is that when $n=1$, you can move the disc directly from $A$ to $B$, without involving $C$ at all, which you feel contradicts the phrase "using $C$", since $C$ isn't actually used. However, in mathematics such phrases are usually inclusive rather than exclusive &mdash; it is not obligatory to actually use $C$. This highlights the fact that mathematical terms often differ in meaning from their common use. For example, when mathematicians say "$p$ or $q$", they are also allowing the possibility "both $p$ and $q$", although common knowledge precludes it.