All three conditions are required, as indicated by the diagram below which contains three surfaces, each satisfying two of the conditions but not the third.
The second surface is the double cone over a triangle where the cone points are skewed vertically. All the local extreme vertices are on the convex hull satisfying (1), but an edge of the convex hull is missing (the one between the left- and right-most vertices of the surface). A horizontal plane can cut off these vertices so the surface does not have the two-piece property and is not tight.
Finally, the third surface has all its local extreme vertices on the convex hull, satisfying (1), and all the edges of the convex hull are part of the surface, satisfying (2), but the apex of the surface is doubly covered, once from the left and once from the right (this is a sphere with two points touching). Again, a horizontal plane can slice off the top, which will cut off two pieces from the rest of the surface, so it is not tight.
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The Geometry Center