Difference of normalization between different definitions of $q$-expansion

We can think of a modular form (say of weight $k$ and level $1$ for simplicity) as a holomorphic function on the upper-half plane $f:\mathbb{H} \longrightarrow \mathbb{C}$ satisfying

$$ f\left( \frac{a\tau + b}{c\tau + d} \right) = (c \tau+d)^k f(\tau) $$ for all $\tau \in \mathbb{H}$, and which is bounded in vertical strips. In particular, we can thing of $f$ as a holomorphic function of $q = \exp(2 \pi i \tau)$ on the punctured open unit disk, which extends at $0$, so it has a $q$-expansion $\sum_{i=0}^\infty a_n q^n$.

One can also think of $f$ as a Katz modular form $F$, the latter we think of as a function on pairs $(E/R,\omega)$ of an elliptic scheme $E$ over a $\mathbb{C}$-algebra $R$ and a nowhere vanishing Kähler differential for the map $E \longmapsto \mathrm{Spec}(R)$. We have $$f(\tau) = F(\mathbb{C}/(\mathbb{Z} \tau + \mathbb{Z}), dz),$$ where we should really replace $(\mathbb{C}/(\mathbb{Z} \tau + \mathbb{Z}), dz)$ by the corresponding elliptic curve and Kähler differential.

The function $F$ has a well-defined notion of $q$-expansion, given by evaluating $F$ at the (base change to $\mathbb{C}$ of) the Tate curve $T(q)$ with its canonical differential. Evaluating at $q = \exp(2 \pi i \tau)$ for $\tau \in \mathbb{H}$, we get a function on the upper-half space. In the case of the classical $q$-expansion, this is just $f$. In the case of the $q$-expansion as a Katz modular form, I think the function should be $$ F((\mathbb{C}/(\mathbb{Z} \tau + \mathbb{Z}), 2 \pi i dz) = (2\pi i)^{-k} f(\tau)$$ (same caveat as above that we should instead write the corresponding elliptic curve and differential) (for example, as can be seen in section A1.2 of Katz's p-adic properties of modular curves and modular forms). This is due to the canonical differential on $\mathbb{C}^*/q^{\mathbb{Z}}$ being $dt/t$ for $t$ the coordinate on $\mathbb{C}^*$, which is $2 \pi i$ times the multiple of the canonical differential $dz$ on $\mathbb{C}/(\mathbb{Z} \tau + \mathbb{Z})$.

In particular, there seems to be a slightly annoying change of normalization between the two definitions of $q$-expansion. For example, it seems that the notion of a modular form defined over a subring of $\mathbb{C}$ should depend on the choice of definition.

Is there a mistake in my reasoning? Am I misunderstanding the correspondence between the different notions of modular forms?