I am a little bit confused about the basic theory of overconvergent modular forms, so here is a question that I think will be straightforward for those who know the theory but would help me a lot.

The question concerns the relationship between various definitions of overconvergent modular forms in the standard papers of Coleman ("Banach spaces and families of modular forms"), Katz ("p-adic properties of ..."), and Chenevier ("une correspondence de Jacquet-Langlands p-adique" https://arxiv.org/abs/math/0301032), where section 3 contains a very brief but useful description of the basic setup).

In section B.2 of Coleman, if $0 \leq v < p^{2-m}/(p+1)$, the definition of the $v$-overconvergent locus in $X_1(Np^m)$ is a little bit complicated: you need to enforce the condition $v(E_{p-1}) \leq v$ as usual, but then you also add some conditions related to the level structure, as follows. For a given point $x$ representing the data $(E, \alpha_N, \alpha_p)$ [the level structure at $p$ being $\alpha_p : \mu_{p^m} \to E[p^\infty]$], in order to include $x$ in the $v$-overconvergent locus, Coleman asks that the image of $\alpha_p|_{\mu_p}$ is the canonical subgroup of order $p$, and that the image of $\alpha_p|_{\mu_{p^{m-1}}}$ is the canonical subgroup of order $p^{m-1}$ (at least this is my interpretation of what is going on in section B.2). On the other hand, in Chenevier, it is simply defined to be the connected component of $\infty$ in the locus where $v(E_{p-1}) \leq v$. Why are these definitions the same ? Is the point that $X_1(Np^m)$ itself might not be connected, or is the point that removing the too supersingular discs makes it disconnected ? Also, what is the relationship between all of this and the definition in Katz of "modular forms with growth condition" as rules defined on tuples of modular data ?