What is the substantial generalization of the “acuteness” to 3-simplex?

Question

A 2-dimensional simplex is just a triangle, and a 3-dimensional simplex is just a tetrahedron… therefore for convenience, I simply use the term triangle and tetrahedron in the following words. We know that an arbitrary (Euclidean) triangle is either right or oblique, and an oblique triangle is either acute or obtuse. In accordance with the remarkable de Gua's theorem, the three-dimensional analog of the right triangle is far and away the trirectangular tetrahedron. However, there does not appear to be an generally acknowledged three-dimensional analogue of the acute triangle as yet; different scholars adopt distinct extensions in their papers. For instance, according to this article, this article, and this article, the following definitions (about a (non-degenerate) tetrahedron) are mentioned:

1. all [of] its dihedral angles are smaller than $\fracπ2$, or the orthogonal projection of each vertex onto the plane of the opposite face lies within that face;
2. it contains its circumcenter in the interior, or its vertices do not lie on a hemisphere of the circumscribed sphere;
3. the circumcenter of each face lies inside the face, or all facets of it are acute;
4. each of its four solid angles' measure is less than $\fracπ2$.

It is noticeable that all angles of a triangle are smaller than $\frac\pi2$ iff its circumcenter lies inside the triangle, and hence both ⒈ and ⒉ can be viewed as practical generalizations of “acuteness” to three dimensions (although the two are not equivalent). Still, analogizing those conditions for being acute triangles, one may consider the following definitions (about a tetrahedron $ABCD$):

5. its Monge point $M$ (the reflection of the circumcenter $O$ in the centroid; alternatively, $\overrightarrow{OM}=\frac{\overrightarrow{OA}+\overrightarrow{OB}+\overrightarrow{OC}+\overrightarrow{OD}}2$) lies in the interior;
6. $S_{\triangle{ABC}}^2+S_{\triangle{ABD}}^2+S_{\triangle{ACD}}^2>S_{\triangle{BCD}}^2$ and three other similar inequalities (cf. the law of cosines for the tetrahedron);
7. ${\left\lvert AB\right\rvert}^{-2}+{\left\lvert AC\right\rvert}^{-2}+{\left\lvert AD\right\rvert}^{-2}>h_A^{-2}$ and three other similar inequalities, where $h_A$ denotes the length of the altitude from vertex $A$;
8. the distance between $O$ and $M$ is less than the tetrahedron's circumradius;
9. et cetera.

All of them (perhaps apart from ⒊) can be viewed as the necessary and sufficient conditions that a tetrahedron is “acute”, but which condition is the most substantial generalization of that for the acute triangle? (Aren't there any convincing criteria?)