Two minor things: In "case 3" there is a typo, it should be $f'=g_i-qf$ I think. Second, I think the condition in "case 2" should instead read "$d$ divides both $\langle e,g_i\rangle$ and $\langle f,g_i\rangle$" (rather than requiring $\langle f,g_i\rangle=0$).

@Zariski93 Yes, I think that description is correct. Here $\eta_n$ now has to be like in D\'efinition 3.1 of Le Halo spectral, with $\alpha=T$.

@Zariski93 Regarding your second comment: Yes, you're right, AIP use T to define the condition on the Hasse invariant. But when you specialise at a point $w$ of weight space, this sends $T\to p^{\epsilon}$ for some $\epsilon$ depending on $w$. This is related to the fact that so far we haven't talked about the relation between the $r$ of $\mathfrak X_{r}$ in Le Halo Spectral and the $\epsilon$ in the torsion paper: The relation depends on the $p$-adic weight.

Once the $\eta$'s are sorted, one can show the alternative moduli interpretation of $\mathfrak X^{\ast}(p^{-n}\epsilon)$ by comparing the two moduli functors: The transformations sending $(E,\alpha,\eta_n)\to (E/C_n,\alpha/C_n,\eta_n,E[p^n]/C_n)$ and $(E,\alpha,\eta_n,D_n)\to (E/D_n,\alpha,\eta_n)$ should define an equivalence.

If we also worry about $\eta$, I think it might not be possible to reconstruct the $\eta$ of $x=(E,\alpha,\eta)$ from $F(x)=(E',\alpha',\eta')$ and $D_n$, at least not over the locus of supersingular reduction. So instead, in order to say what "$\mathfrak X^{\ast}_{\Gamma_0(p^n)}(\epsilon)_a$" is (i.e. the alternative moduli interpretation) one might need to talk about tuples of the form $(E,\alpha,\eta,D_n)$ where $\eta$ is defined in terms of $E/D_n$. I've edited my post to reflect this.

@A.Walker: In answer to your question, I should perhaps say that in all of the above I was focusing on $(E,\alpha,\eta)$ etc and I'll admit I swept the $\eta$'s under the carpet. Sorry -- the post was so long already :). The $\eta$'s make things slightly more complicated, also because they are defined slightly differently for Scholze and Andreatta--Iovita--Pilloni (one uses $\eta Ha^{p^{r+1}}=p$, one $\eta Ha=p^{\epsilon}$, this is a normalisation issue), but to describe the moduli functor, we should of course include them. I didn't mean to imply that you can reconstruct $\eta$ from $D_n$.

Thanks! @Zarisiki93 Regarding your first comment, the things is that a priori we didn't define a formal model of $\mathcal X^{\ast}_{\Gamma_0(p^n)}(\epsilon)_a$, but of $\mathcal X^{\ast}(p^{-n}\epsilon)$. The Atkin-Lehner isomorphism is then used to show that the formal model of the latter is also a formal model of the former. Re moduli interpretations of this: