There seems to be a confusion between two different aspects of a lens.
A converging lens takes a parallel beam of light and brings it to a focus at a point. It is generally thicker at the centre than at the edge. Examples of this are: a magnifying glass, and eyeglasses for an individual who is farsighted and cannot focus on nearby objects.
A diverging lens takes a parallel beam of light and spreads it out, as if it were coming from a point. Such a lens could be used for glasses for a near-sighted person who cannot focus on distant objects
None of the above says anything definitive about the shape of the each of the two surfaces of the lens; only about the relative thickness at the centre and edge.
- Convex-convex - always converging; always thicker at the centre
- Plano-convex - always converging; always thicker at the centre
- Concavo-concave - always diverging; always thinner at the centre
- Plano-concave - always diverging; always thinner at the centre
- Convex-concave - can be converging, diverging or neutral
All these lenses will behave the same way, no matter which way the light goes through them.
Eyeglass lenses will almost always be convex on the outer surface, the one farthest from the eye, simply to fit it to the curvature of the face.
If the inner surface is concave, and more sharply curved than the outer, then the lens is diverging.
If the inner surface is more gently concave, flat, or convex, then the lens is a converging one.
Stage glasses, as part of the costume for an actor with normal vision, would have a convex outer surface and a concave inner surface, with the same radius of curvature for both surfaces, and would be neither converging or diverging...