Geometry of the $q$-expansions of Katz modular forms

Let $N \geq 5$ be an integer so that the $\Gamma_1(N)$-moduli problem is representable over $\mathbb{Z}[1/N]$ (both in terms of elliptic curves/generalized elliptic curves). I am interested in Katz modular forms of this level and their $q$-expansions. From my modest understanding, there are two ways to define $q$-expansions :

1. Let $f$ be a "meromorphic" Katz modular form (say defined over $\mathbb{Z}$), i.e a rule which assigns to a triple $(E/R,t,w)$ (where $E/R$ is an elliptic curve over a ring, $w$ a trivialization of its Hodge bundle and $t$ a level structure) a scalar in $R$ + compatibility with base change and weight k homogeneity. Then one can evaluate $f$ at triples consisting of the Tate curve, its canonical differential and some level structure. This will yield some finite tailed Laurent series.

Holomorphic forms then correspond to those forms with holomorphic expansions, i.e their expansions land in some ring of formal power series. Now one can show that such rules correspond to global sections of the Hodge bundle $\omega ^{\otimes k}$ on $Y_1(N)$. This bundle has an extension that I will still denote by $\omega$, to the complete modular curve $X_1(N)$. So meromorphic modular forms can be seen as meromorphic sections of $\omega^{\otimes k}$, holomorphic on the open affine modular curve $Y_1(N)$.

2. Let $f \in H^0(X_1(N),\omega^{\otimes k})$, for the sake of implicity let us consider the cusp $\infty$. This is a closed immersion $\mathbb{Z}[1/N,\zeta_N] \to X_1(N)$. The formal completion of $X_1(N)$ along $\infty$ is identified with $\mathrm{Spf} \mathbb Z[1/N,\zeta_n][[q]]$ via the $\mathbb Z[1/N,\zeta_n][[q]]$-point of $X_1(N)$ given by the Tate curve (seen as a generalized elliptic curve) + the trivialization ($w_{\mathrm{can}}$) of the pushforward of its relative dualizing sheaf (call it $\omega_{\mathrm{Tate}}$) that restricts to the canonical differential on $\mathbb Z[1/N,\zeta_n]((q))$ and the $\Gamma_1(N)$-structure given by $\zeta_N$. Now the global section $f$ gives rise to a global section of the completion of $\omega^{\otimes k}$ along $\infty$ which in turn (using some base-change compatibilities) yields by pulling back along the previous isomorphism, some global section of the completion of $\omega_{\mathrm{Tate}}^{\otimes k}$ which is trivialized by $w_{\mathrm{can}}^{\otimes k}$. So formally along $\infty$, $f$ pullbacks to some $g(q)w_{\mathrm{can}}^{\otimes k}$ with $g(q)$ some formal power series.

The second construction is of a more geometric flavour than the first one but it works for sections defined over the whole of $X_1(N)$. However the first construction is well defined even for meromorphic modular forms. I guess there should be some geometric way to define $q$-expansions for meromorphic modular forms, in the same spirit as the second construction which would amount to think of meromorphic modular forms as holomorphic sections over punctured neighbourhoods of the cusps, pulling them back in some formal punctured neighbourhood (whatever that means) and getting some expansion of the form $h(q) \times w_{\mathrm{can}}^{\otimes k}$ with $h(q)$ some Laurent series (as one would expect from taking a punctured neighbordhood). Also, a meromorphic modular form should be holomorphic iff its expansions at punctured neighbourhoods are given by formal power series.

However, because I am not very familiar with formal schemes, I do not know how to formulate this precisely, nor have I found a reference discussing it in this context.

Is this picture correct? If so, how should the punctured (formal) neighborhood and the corresponding Laurent expansion be defined? References would also be very welcome.