Suppose that $R$ is a complete DVR with characteristic 0 fraction field $K$, maximal ideal generated by $p$ and characteristic $p>0$ residue field $k$ which is algebraically closed. Suppose that $\mathcal{E}$ (proper and a group scheme over $R$) is a model of an elliptic curve $E/K$ and denote by $E_{0}$ the special fibre over $k$. We assume that $E$ has good ordinary reduction at $p$ so that $p$-torsion of $E_{0}$ is given by a split exact (connected-etale) sequence of finite flat $k$-group schemes:
$$ (*)\quad 0 \rightarrow \mu_{p} \rightarrow E_{0}[p] \rightarrow \mathbb{Z}/p\mathbb{Z} \rightarrow 0$$
I would like to understand the possibilities of the finite group scheme $\mathcal{E}[p]$ of $p$-torsion of $\mathcal{E}$. I believe this should be equivalent to asking what possible lifts exist of $(*)$. Further, I believe (perhaps by uniqueness of lifts of etale group schemes and some form of duality) the connected-etale sequence should still have the form:
$$0 \rightarrow \mu_{p,R} \rightarrow \mathcal{E}[p] \rightarrow \mathbb{Z}/p\mathbb{Z} \rightarrow 0$$
now considering the outside groups as $R$-schemes. In this sense, understanding the liftings should be equivalent to understanding the possible extensions of $R$-group schemes above. In this paper by Rene Schoof: https://link.springer.com/content/pdf/10.1007/s00229-011-0509-y.pdf it appears that he classifies all possible extensions as being group schemes of the form: $$G_{\epsilon}=\text{Spec}\left(\bigoplus_{i=0}^{p-1}\frac{R[X_{i}]}{(X_{i}^{p}-\epsilon^{i})}\right) $$ where $\epsilon \in R^{*}$.
I would like to know if the above argument is correct and this indeed gives all possible lifts of the $p$-torsion of $E_{0}$ and more importantly (as a softer question) is there any geometric meaning I can assign to $\epsilon$? Namely, if I am given even the Weierstrass equation of an elliptic curve $E$, are there any geometric invariants/properties of $E$ which will tell me what $\epsilon$ is without necessarily finding a model of $E$ and reducing?
It would also interesting to know, failing the above if there is some criteria on $E$ to know when the lifted exact sequence is still split (perhaps with some relation to Serre-Tate theory and a 'trivial' lifting of the special fibre).
