An octahedron shape is one which has 8 faces. Since it is a prism it must be a hexagonal prism (2 hexagons and 6 rectangles = 8 faces). And a hexagonal prism has 2*6 = 12 vertices.
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A truncated octahedron has 24 vertices.
An equilateral triangle based anti prism is simply a "normal" octahedron, as described by someone trying to make things sound more complicated than they are. It has 6 vertices.
It depends on the exact shape. An octahedron is a shape with eight faces. It can be a heptagonal pyramid (8 vertices, 14 edges), or a hexagonal prism (12 vertices, 18 edges), square dipyramid (6 vertices, 12 edges) are some examples. There are more.
An octahedron has eight faces. If it has 62 faces then it is not an octahedron!
An octahedron is the name for a non-specific shape. It could be a di-rectangular pyramid with 6 vertices - this is like two rectangular [or square] based pyramids stuck together base-to-base. Or it could be a heptagonal pyramid, in which case it has 8 vertices. Or it could be a hexagonal prism, in which case it has 12 vertices. There are other octahedraons.