Are there different words for a circle, and the edge of a circle, which are topologically distinct? The following shape, we would refer to as a circle:
First circle
The following shape we would also refer to as a circle:
Second circle
But these two circles are topologically distinct from one another, are they not? The first circle has a massive "hole" in the middle, and so is really more of a loop in two dimensions. The second circle is a "true circle". But we would refer to both as a circle. In fact, the wikipedia page on circles shows images that mirror the first circle: https://en.wikipedia.org/wiki/Circle
But the first circle is less of a circle object than the second - it's really a loop holding a circular form, or the edge of a circle, or a circle with a hole punched in it, than a circle.
I was thinking about this because of Nietzsche's quote: "time is a flat circle". Would he mean to say that time is a normal, two dimensional circle, akin to the second picture of a circle above? Or that time is a loop, like a piece of flat ribbon, folded back on itself? I tend towards interpreting it as the second option, as that makes more sense: he's saying that in the end everything repeats and there are no beginnings nor endings, just an eternal cycle. But that's more metaphysical, the specific question for this post is whether there are different words for these two, clearly topologically distinct, 2D objects, which we refer to as circles?
 A: I would call the first figure (the curve) a circle and the second one (the area) a disc.
A: In addition to the answers given on "circle" being the boundary of a "disk" in a 2-dimensional plane: In arbitrary dimensions one usually calls the set of all $x$ in $\mathbb R^n$ with $\|x\|\le 1$ the (closed) $n$-dimensional unit ball and its boundary, the set of all $x$ with $\|x\|=1$, the $(n-1)$-dimensional unit sphere. Here $\|x\|$ denotes the distance from the origin.
So the first figure would be a $1$-dimensional sphere and the second a $2$-dimensional (closed) ball.
The names have their origin in the case $n=3$ where the $3$-ball actually is a solid ball as you think of it and the $2$-sphere is just the surface of the ball.
A: A circle is the set of points in the plane that all have a fixed distance of exactly $r>0$ to a given centre point. So your second set is a circle plus its interior area, i.e. all points that have distance $\le r$ to that same centre. Such a set is called a (closed) disk. It's two-dimensional as opposed to the one-dimensional circle (topologically we can assign a dimension (integer) to all subsets of the plane or more generally a separable metric space). So for sure they are topologically distinct: we can also observe that removing two points from a circle leaves us with two pieces (a disconnected set), while in a disk we can remove any two points and the remainder is still connected (i.e. in one piece). This observation is related to the fact of their differing dimensions, in fact.
I would leave Nietzsche out of a maths discussion, personally.
