Help me settle a family debate.
We’re trying to figure out what the word “round” means. Some say that ovals and circles are both round. Others believe that ovals are rounded (though not round), and only circles are round. Your thoughts?
Roundness is a mathematical idea, so it really comes down to what you mean by the word. I’d say that any infinitely differentiable simple closed curve with convex interior is round, so yes, ovals qualify.
Not saying your family’s choice of topics isn’t great, Elendil’s Heir, but I think this is more of an IMHO thing. Moved GD -> IMHO.
“Rounded” being different from “round” strikes me as a desperate reach of an argument. Ovals are round because they don’t have angles. A baseball helmet is round. A dome is round. My belly is (sadly) round.
I have to disagree. To me, “round” means circular, spherical or cylindrical . . . or any part thereof. “Rounded” means any other type of curve.
Baseballs and Bats are round. Footballs are not. Round means circular or spherical. Wiki agrees, as does the dictionary.
Round is not a mathematical idea. There is nothing in calculus that describes anything “round”, curved yes, but not round. The only possible “mathematical” definition of round has to do with decimals and that has nothing to do with the OP.
Ovals are ovoid, ellipses are elliptical, circles are circular (aka round).
Many of those dictionary usage items seem to be talking about round in the sense around.
Other dictionaries include broader definitions, such as freedom from angularity. Round Definition & Meaning | Dictionary.com
So just as ‘square’ doesn’t necessarily have to mean ‘a square’, I would consider round to mean ‘having roundness’ - not necessarily being exactly circular - we already have the specific term ‘circular’ for that purpose.
Rounded, on the other hand, implies the addition of roundness - a rounded square or triangle, for example, being a shape resembling a square or triangle, but with roundness added at the corners or on the sides.
There’s circular, ovoid, and oblong.
All are differing degrees of roundositousness.
If a bat is round, due to its circular cross-section, why isn’t a football, for the same reason?
It doesn’t?
As an adjective, it can mean any right angle.
But to say an object, say a box, is square can mean only one thing. Right?
I’d say it has a very loose meaning and can cover any shape/aspect whose overall defining characteristic is curved; either 2D or 3D.
Ovals and circles would be round. Same with Spheres and 3D shapes that have a circular or ovoid cross-section (think a football or a cylinder).
Also, I would describe something like the Epcot Center round, even though it’s comprised of thousands of faced angles/triangular polygons.
*that should read faceted.
I think “square” could reasonably be used to describe shapes that are right angled but not equal in height and breadth. For example, if a doorway of average dimensions had an obvious lean to it, you might expect a carpenter to say it wasn’t square, or was out of square. This is confusing, yes, but not out of the bounds of common usage. Hell, the carpenter might pull a tool called a “square” out of his truck to check it, and said tool would not have any dimension on it appear twice.
I make rounds, but I am not sure my experience is germane to the discussion.
I would have to go with round is not equal to circle or sphere but includes them. Thus “That oval tower has round walls inside” would make sense to me.
A box is not “square” because a box is a rectangular prism, which may have one set of parallel sides that are square, two sets that are square, or three sets. In the last case, we would call it cubical, not square. 
According to Merriam-Webster Online Dictionary,
So it is wrong to assert that “the dictionary” says that round = spherical/circular. That is one usage, but it has other usages. Since there is no precise mathematical term “round,” one cannot resort to such a definition to settle the argument.
As for “rounded,” that would to me imply something that isn’t round by nature having been made round. Thus, if you take a cube and round off the corners, that’s “rounded.”