At least in the ordinary case, the argument is simple so let me recall it here.

The $p$-divisible group $J$ is an extension of an étale $p$-divisible group $J^{et}$ by a multiplicative $p$-divisible group $J^{m}$ and the Pontryagin dual $J^{et*}$ of $J^{et}$ is a free $\mathbb T_\mathfrak{m}$-module of rank 1 by the ordinarity assumption. Assume in addition that $\mathbb T_\mathfrak{m}$ is a Gorenstein ring. Then $\operatorname{Hom}(\mathbb T_\mathfrak{m},\mathbb Z_{p})/\mathfrak{m}\operatorname{Hom}(\mathbb T_\mathfrak{m},\mathbb Z_{p})$ is a $\mathbb T_\mathfrak{m}/\mathfrak{m}\mathbb T_\mathfrak{m}$-vector space of dimension 1 (here we use the fact that a ring $R$ is a Gorenstein ring if and only if its dualizing complex is concentrated in degree 0 and isomorphic to $R$).Then $J^{et}[\mathfrak m]$ is free of rank 1, by duality so is $J^{m*}[\mathfrak m]$ and finally $J[\mathfrak m]$ is a free $\mathbb T_\mathfrak{m}$-module of rank 2.

Note that this does not require the fact that $\bar{\rho}_{\mathfrak m}$ is absolutely irreducible.

In the ordinary case, the argument is simple so let me recall it here.

The $p$-divisible group $J$ is an extension of an étale $p$-divisible group $J^{et}$ by a multiplicative $p$-divisible group $J^{m}$ and the Pontryagin dual $J^{et*}$ of $J^{et}$ is a free $\mathbb T_\mathfrak{m}$-module of rank 1 by the ordinarity assumption. Assume in addition that $\mathbb T_\mathfrak{m}$ is a Gorenstein ring. Then $\operatorname{Hom}(\mathbb T_\mathfrak{m},\mathbb Z_{p})/\mathfrak{m}\operatorname{Hom}(\mathbb T_\mathfrak{m},\mathbb Z_{p})$ is a $\mathbb T_\mathfrak{m}/\mathfrak{m}\mathbb T_\mathfrak{m}$-vector space of dimension 1 (here we use the fact that a ring $R$ is a Gorenstein ring if and only if its dualizing complex is concentrated in degree 0 and isomorphic to $R$).Then $J^{et}[\mathfrak m]$ is free of rank 1, by duality so is $J^{m*}[\mathfrak m]$ and finally $J[\mathfrak m]$ is a free $\mathbb T_\mathfrak{m}$-module of rank 2.

Note that this does not require the fact that $\bar{\rho}_{\mathfrak m}$ is absolutely irreducible.

If $T_p$ belongs to $\mathfrak m$, the argument is more involved but Proposition (14.2) of Modular curves and the Eisenstein Ideal asserts that $J[\mathfrak m]$ is always of dimension 2 in that case (even without assuming $\mathbb T_\mathfrak{m}$ to be Gorenstein).

So the question of the title definitely admits the answer yes and conversely for the question in the body of the text.