I’m probably the wrong person to best describe the various figuring terms, but I think some of the confusion may come from the whole reflector vs. refractor thing. And then there is ambiguity in “smooth” as well!
For a Newtonian reflector if you have a paraboloid figure to the mirror (figure meaning the shape of the cross section of the mirror) you will have no SA. When you grind a mirror it will form a spherical surface naturally with no effort - if you think about it if you rub two blanks together they can *only* form a spherical shape between them.
Once you have a spherical surface of the right focal length it must be adjusted to form a paraboloid. This means removing material from the outer edge and requires different polishing strokes that preferentially remove material from the outer edge. This is very challenging to get right. You might remove too much from say 50-75% of the way to the edge and not enough from 75% out to the edge. This will result in SA because two different “zones” (donut shaped regions centered on the mirror) now focus light at a different focal length.
I think “smooth figure” usually means that the surface properly follows the paraboloid shape but often the result is somewhat wavy. A key point though is that a sphere is not the correct shape to begin with though! If you make a perfectly smooth spherical mirror you will have SA.
Now the above is again all for reflectors which is probably a distraction in a document for refractors! So I’d drop the whole idea of smooth and figuring as that really doesn’t apply to refractors at all.
As far as I know all amateur refractors use spherical lenses. Aspheres are common in camera lenses these days because we can economically make small aspheric elements but telescope objectives are far too large to do that economically.
The way SA is dealt with in a refractor that only has spherical surfaces is by combining multiple spherical surfaces of different radii. In a doublet there are four spherical surfaces (two on each element). The focal length actually doesn’t constrain any one of those surfaces! It to a first order only constrains the relationship between the front most surface and the rear most surface. So you can in theory choose *any* radius of curvature for the front surface of a doublet and your chosen focal length will the tell you what the rear surface curvature must be. Completely unconstrained are the inner surfaces of the doublet. You can change those however you want and not effect the focal length of the doublet. And then there is the spacing of the two elements. And then the thickness of the two elements. And the index of refraction for the two different glasses...
The key thing to understand is that in a refractor for a given focal length there are multiple spherical shapes of the lens elements that will give the same focal length. Depending on which shape you choose you will get more or less of a particular aberration - SA being just one.
This shows how the shape of a single element lens optimizes for SA: http://hyperphysics....eoopt/aber.html
With a Newtonian reflector there is only one “correct” shape for a given focal length and F-ratio. That is a paraboloid and it will have no SA and no chromatic aberration. It will have coma and there is nothing you can do about it with the mirror shape. For a Newtonian once you set the focal length and F-ratio there are absolutely no choices left for the designer.
As described for a doublet there are a host of choices that allow the designer to trade one aberration against another. There is no “correct” design since one can decide they want to optimize on axis SA at the expense of off axis coma or they can choose the opposite depending on the application or desired use.
I’m not sure that was so illuminating... let me try this summary:
In a Newtonian SA is the result of a manufacturing *error* - the mirror is the wrong shape and perhaps this could be called a non-smooth figure.
In a refractor SA is usually the result of a design *choice* - the designer must balance multiple aberrations against each other in the design and if they over optimize SA they may end up with worse performance due to a different aberration (coma, astigmatism, field curvature).
Obviously if someone screws up manufacturing a refractor that could also introduce even more SA than should be there! But that isn’t the typical source of SA in a refractor, rather it is the result of design choices.