Given integers $m,n$ and $c \geq 2$, the Kloosterman sum is defined as $S(m,n;c) = \sum_{k \in (\mathbb{Z}/c\mathbb{Z})^{\times}}{e^{\frac{2i\pi}{c}(mk+nk^{-1})}}$, where $k^{-1}$ is the reciprocal of $k$ mod $c$.
The following properties are easy to compute (and some are listed on the Wikipedia page):
1) If $q$ is a power of the prime $p$, $pS(m,n;q)=S(pm,pn;pq)$.
2) If $ua+vb=1$ with $c=ab$, $S(m,n;c)=S(vm,vn;a)S(um,un;b)$.
Now, we have from e.g. Iwaniec & Kowalski, Analytic Number Theory (Cor. 11.12), $|S(m,n;c)|\leq \tau(c)(m,n,c)^{1/2}c^{1/2}$ where $(m,n,c)$ is the gcd of $m,n,c$ and $\tau$ is the divisor-counting function.
The proof is as follows:
1) If $p$ is prime, $m$ and $n$ are coprime with $p$, $|S(m,n;p)| \leq 2\sqrt{p}$ (the most involved part of the proof, by Weil).
2) If $p$ is an odd prime power, Salie identities (from Wikipedia, or Chapter 12 of the same book) yield, for integers $m,n$ coprime to $p$, $|S(m,n;p)| \leq 2\sqrt{p}$ as well.
3) We combine 1) and 2) with the multiplicativity properties above to show the result in the general case.
My question is the following: here, we see a contribution of $2$ for each odd prime factor of $c$. So why, in the final bound, is there a divisor-counting function appearing? The logical explanation is that $S(m,n;2^l)$ gives a bound along the lines of $(l+1)2^{l/2}$, but is that the only reason why we use the divisor function instead of some more ad hoc arithmetic function in the bound?
Additionally, that case $c=2^l$ does not seem specifically explained in the book, and the argument does not seem to involve the exponent, so that the same bound as for odd prime powers seems to hold. So what happens for powers of $2$? Does anyone know of a reference where this is addressed?
TL;DR: Why does one use the divisor-counting function in the bounds for Kloosterman sums, instead of, say, $\tau'(n)=(1+v_2(n))\prod_{p|n,p\text{ odd}}{2}$ which should work by multipicativity and Selie identities? Would the even smaller $\tau''(n)=\prod_{p|n}{2}$ work?