What is the definition of Breuil-Kisin module ? Does the definition varies author to author ?

**Notations:** Consider a finite extension $K supset mathbb{Q}_p$ of the $p$-adic field, with ring of integers $mathcal{O}_K$. Let $pi in mathcal{O}_K$ be the uniformizer and let $kappa$ be the residue field of $K$. Also let, $E(u)$ be the Eisenstein (minimal) polynomial of the uniformizer $pi$ and denote $mathfrak{S}:=W(kappa)((u))$, where $W(kappa)$ is the ring of Witt vectors on the residue field $kappa$. Let $varphi$ be the Frobenius map which extends to $mathfrak{S}$ from $W(kappa)$, which itself an extension of the Frobenius on $kappa$. Finally let $G$ be a $p$-divisible group on $mathcal{O}_K$ and denote by $BT(mathcal{O}_K)$

**$underline{Definition}$ 1:** We define the by $BT_{/mathfrak{S}}^{varphi}$ the category of finite free $mathfrak{S}:=W(kappa)((u))$-module $mathfrak{M}$ equipped with an injective map $varphi: mathfrak{M} to mathfrak{M}$ such that the cokernel of its $mathfrak{S}$-linearisation $1 otimes varphi: mathfrak{S} otimes_{mathfrak{S}} mathfrak{M} to mathfrak{M}$ is killed by $E(u)$.

It is famous result conjectured by Breuil and proved by Kisin, that the categories $BT(mathcal{O}_K)$ and $BT_{/mathfrak{S}}^{varphi}$ are equivalent.

Can we say that the **Defintion 1** defines Breuil-Kisin modules or at least belongs to category of Breuil-Kisin modules ?

Though I have several definitions of Breuil-Kisin modules in several articles, some of them assuming finite generated $mathfrak{S}$-modules and some of them assuming finite free $mathfrak{S}$-modules, as follows:

**$underline{Definition 2:}$** In this article by Gee (here $mathfrak{M}=M$),

Almost same definition in this article by Hui Gao, as follows:

**$underline{Definition}$ 3:** Finally, in page 5-6 of this article by Bryden Cais, the Breuil-Kisin module is as follows (assuming finite free):

**My Question/Request:**

$(1)$ Can we say that the **Defintion 1** defines Breuil-Kisin modules or at least belongs to category of Breuil-Kisin modules ?

$(2)$ Definition $2$ assumes finitely generated property while Definition $3$ assumes finite free property. How are they same ?

$(3)$ Is the Breuil-Kisin module any module defined over the ring $mathfrak{S}:=W(kappa)((u))$ ?

$(4)$ Does the definition of Breuil-Kisin module varies author to author ? If so, what is the most common definition of Breuil-Kisin module ?

Any comments on the above $4$ questions are appreciated.