So I would like to consider a lattice $\Lambda \subseteq \mathbb{C}$ generated by $(1,\tau)$ with $\tau$ in the upper-half complex plane. This lattice $\Lambda$ will remain fixed. If you choose the fundamental parallelogram to have vertices $\{0, 1, \tau , \tau+1\}$, then under the modular transformation $\tau \to \tau+1$, this fundamental domain essentially just "stretches" horizontally. It will then have vertices $\{0, 1, \tau+1, \tau+2\}$.
What I've been confused about, is how to generate similar "vertical stretches." What I mean is transforming the initial fundamental domain to a scaled one with vertices $\{0, \tau, \tau+1, 2\tau + 1\}$.
I had assumed it would be some composition of $\tau \to \tau+1$ and $\tau \to -1/\tau$, but this doesn't seem to work. I think these "stretches" are called Dehn Twists. By symmetry, I can't fathom why the horizontal stretches are allowed but the vertical ones not. After all, you're still using the same lattice, you're just choosing a different fundamental domain. Any tips on what I'm overlooking here?
It seems to me that your confusion is rooted in misunderstanding the proper roles of two relevant actions of the modular group $SL(2,\mathbb{Z})$:
Action #1: $SL(2,\mathbb{Z})$ acts on $\mathbb{R}^2$ by linear transformations. This is used to obtain an isomorphism between $SL(2,\mathbb{Z})$ and the mapping class group of the torus (more below).
Action #2: $SL(2,\mathbb{Z})$ acts on the upper half plane of $\mathbb{C}$ by fractional linear transformation. This is used to obtain an action of the mapping class group of the torus on the Teichmuller space of the torus which is identified as the upper half plane of $\mathbb{C}$ and which can be thought of as the space of normalized Euclidean structures on the torus via fundamental parallelograms.
If your desire is to understand how certain matrices correspond to Dehn twists, you need to focus on Action #1 and ignore Action #2. So I'm going to ignore your variable $\tau$, which is associated to Action #2, and which has nothing to do with understanding a Dehn twist on the torus.
I'm going to fix the lattice $\Lambda$ to be generated by $\begin{pmatrix}1 \\ 0 \end{pmatrix}$ and $\begin{pmatrix}0 \\ 1 \end{pmatrix}$.
The quotient torus is $T = \mathbb{R}^2 / \Lambda$.
Associated to each $M = \begin{pmatrix}a & b \\ c & d \end{pmatrix} \in SL(2,\mathbb{Z})$ I have the linear transformation $$M \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix}ax+by \\ cx+dy \end{pmatrix} $$ This linear transformation preserves the lattice $\Lambda$ and all its orbits under the additive action of $\Lambda$ on $\mathbb{R}^2$, and it therefore descends to a homeomorphism of the quotient torus $T$. Thus to each $M \in SL(2,\mathbb{Z})$ we have associated a homeomorphism of $T$, and this defines the isomorphism $SL(2,\mathbb{Z}) \approx \text{MCG}(T)$.
Let's consider two elements of $SL(2,\mathbb{Z})$ namely $$A = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix} \quad\text{and}\quad B = \begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix} $$ The linear action of $A$ is a "shearing" transformation (not a "stretching" transformation) given by the formula $$A\begin{pmatrix}x \\ y \end{pmatrix} = \begin{pmatrix}x+y \\ y \end{pmatrix}$$ This linear transformation preserves lattice $\Lambda$ and descends to a "horizontal" Dehn twist on $T$. Similarly, $B$ preserves $\Lambda$ and descends to a "vertical" Dehn twist on $T$.