Milne's Modular Functions and Modular Forms states, at the bottom of page 10:
We can normalize our lattices so they are of the form $$\Lambda(\tau) := \mathbb{Z} \cdot 1 + \mathbb{Z} \cdot \tau$$ for some $\tau \in \mathbb{H}$. Note that $\Lambda(\tau) = \Lambda(\tau')$ if and only if there is a matrix $$\begin{pmatrix} a & b \\ c & d \end{pmatrix} \in {\rm SL}_2(\mathbb{Z})$$ such that $$\tau' = \frac{a \tau + b}{c \tau + d}.$$
I can't make sense of this last sentence. For instance, if we take the matrix $$\begin{pmatrix} 1 & 0 \\ 2 & 1 \end{pmatrix} \in {\rm SL}_2(\mathbb{Z})$$ then we have $$\begin{pmatrix} 1 & 0 \\ 2 & 1 \end{pmatrix} \cdot i = \frac{i}{2 i + 1} = \frac{i(2i - 1)}{(2i)^2 - 1^2} = \frac{-2 - i}{-5} = \frac{2}{5} + \frac{1}{5} i$$ and this is clearly not an element of $\Lambda(i) = \mathbb{Z} + \mathbb{Z} i$.
I think "equals" should be replaced with "is homothetic to." The surrounding context makes it clear that Milne only wants to consider lattices up to homothety anyway, since those describe elliptic curves.