2. It actually looks more like a redefinition than a new discovery: "It may be confusing because Goldberg called them polyhedra, a perfectly sensible name to a graph theorist, but to a geometer, polyhedra require planar faces "
I don't think that's quite right. They narrowed the definition to strict polyhedral, which hadn't been done before. Then showed that they existed.
"Schein and his colleague James Gayed have described that a fourth class of convex polyhedra, which given Goldberg’s influence they want to call Goldberg polyhedra, even at the cost of confusing others. "
Hey! There are in fact infinite solution. Each regular face of an icosahedron for instance can be 'inflated' to form a slight dome, made out of smaller regular polygons.
Each surface polygon is flat. They can be 'inflated' via the OPs technique without violating the bound of an enclosing sphere, right? Each recursive expansion has an inflation factor that scales. Hm. But the sphereical section bounding each polygon doesn't scale, it becomes 'flatter' as you recurse. So there's a limit.
Actually, not. The definition of convex is that given a point A and a point B and a line between A and B, all points on the line AB are in the interior space of the solid.
Inflating two adjacent surfaces creates a valley along the pre-existing edge between the two of them and fails the above definition.
As far as I can tell, the discovery here (if there is one at all) is a method for constructing those polyhedra and others like them and being sure they're actually polyhedral (no curved or bent faces).