When are these definitions of "toric variety" equivalent?

Question

Let $k$ be an algebraically closed field. Let $X$ be an integral $k$-scheme, separated and of finite type over $k$. Let $d := \dim X$, let $T := (\mathbb{G}_{m,k})^{d}$ be the $d$-dimensional torus, and suppose we have an action of $T$ on $X$ over $k$ given by the action morphism $\sigma : T \times_{k} X \to X$. Let $$ \mathrm{stab}_{\sigma} : Z_{\sigma} \to X $$ be the stabilizer of $\sigma$, namely the pullback of the morphism $(\sigma,p_{2}) : T \times_{k} X \to X \times_{k} X$ via the diagonal $\Delta_{X/k} : X \to X \times_{k} X$. For a $k$-point $x \in X(k)$, let $$ \alpha_{x} : T \to X $$ denote the orbit morphism of $x$, namely the composition $\mathrm{id}_{T} \times x : T \to T \times_{k} X$ with $\sigma$, and we denote by $T \cdot x \subset X$ the (set-theoretic) image of $\alpha_{x}$. By the closed orbit lemma, we have that $T \cdot x$ is a locally closed subset of $X$.

Depending on the source, the scheme $X$ is called a toric variety if it satisfies one of the following conditions (ordered roughly from strongest to weakest):

1. The scheme $X$ is given by a construction involving fans (e.g. [4, Theorem 1.4] or [1, Section 2.1]).

2. There is a $T$-equivariant open immersion $j : T \to X$, where $T$ acts on itself by left multiplication.

3. The action $\sigma$ has generically trivial stabilizers, i.e. there exists a dense open subscheme $U \subseteq X$ such that the restriction $\mathrm{stab}_{\sigma}^{-1}(U) \to U$ is an isomorphism.

4. The action $\sigma$ has generically finite stabilizers, i.e. there exists a dense open subscheme $U \subseteq X$ such that the restriction $\mathrm{stab}_{\sigma}^{-1}(U) \to U$ is a quasi-finite morphism.

5. There is a $k$-point $x \in X(k)$ such that $T \cdot x$ is an open subset of $X$.

6. The action $\sigma$ is faithful, i.e. the corresponding morphism of group sheaves $T \to \underline{\mathrm{Aut}}_{(\mathrm{Sch}/k)}(X)$ on the category of $k$-schemes $(\mathrm{Sch}/k)$ is injective.

Do the above conditions imply any others (possibly with mild assumptions about $X$)?

At the moment, I am most interested in whether (4) implies (2).

Here is what I have so far: In (1), the scheme $X$ is obtained by equivariantly gluing certain affine $T$-schemes which satisfy (2), hence (1) implies (2). If $X$ is normal, then (2) implies (1) by Sumihiro's theorem [4, Theorem 1.5]. For (2) implies (3), take any $k$-point $x \in X(k)$ lying in the image of $j$. Clearly (3) implies (4). For the equivalence of (4) and (5), use the orbit-stabilizer theorem for algebraic groups (i.e. that $\dim (T \cdot x) + \dim (x^{-1}(\mathrm{stab}_{\sigma})) = \dim T$, see [2, Proposition 3.20]) and the assumption that $T$ and $X$ have the same dimension (since $T \cdot x$ is locally closed, saying that $T \cdot x$ is open is the same as saying that $\dim (T \cdot x) = d$). For (2) implies (5), either compose previous implications or take $U = j(T)$. For (2) implies (6), restriction via $j$ gives a map $\underline{\mathrm{Aut}}_{(\mathrm{Sch}/k)}(X) \to \underline{\mathrm{Aut}}_{(\mathrm{Sch}/k)}(T)$ such that the composition $T \to \underline{\mathrm{Aut}}_{(\mathrm{Sch}/k)}(X) \to \underline{\mathrm{Aut}}_{(\mathrm{Sch}/k)}(T)$ is injective, hence the first arrow is injective.

References:

[1] Elizondo, Lima-Filho, Sottile, Teitler, "Arithmetic Toric Varieties", Mathematische Nachrichten, vol. 287, no. 2-3 (2014) link

[2] Hoskins, "Moduli problems and geometric invariant theory", online course notes (2015)

[3] Kempf, Knudsen, Mumford, Saint-Donat, "Toroidal Embeddings 1", Springer Lecture Notes in Mathematics, vol. 339 (1973)

[4] Oda, "Convex Bodies and Algebraic Geometry", Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 3 (1985)

Answer 1

As noticed by Dave Anderson, (4) does not imply (2) because the generic stabilizer might be finite but non-trivial. But still, let $E$ be the stabilzer subgroup scheme of $x$ as in (5). Because $T$ is abelian, $E$ is normal in $T$. Hence $E$ acts trivially on the open set $Tx$. The fixed point scheme of $E$ being closed this implies that $E$ acts trivially on all of $X$. Hence the action of $T$ on $X$ factors through the torus $T_0=T/E$ for which then (2) is satisfied.

More interesting is the relationship of (6) and the other conditions. It is known that an effective torus action has trivial generic stabilizers. Therefore (6) implies the other conditions except for (1) where one needs normality.

To see the claim one can e.g. use that subgroup schemes of a torus are rigid. More precisely, let $U\subseteq X$ be open, dense over which $Z_\sigma$ is flat. Then one can show that it is a constant subgroup scheme of $T\times U$. Therefore any fiber $E$ acts trivially on $U$, hence on $X$. Thus $E=1$ by (6).