What is "statistics"?

I've been thinking about what "statistics" means. I came up with a tentative definition on my own, and then a little googling led me to think I'm not entirely wrong.

My intuition:

We take a manifold. We select a point on the manifold as the origin. We "populate" that point with an infinite set, say, the reals between 0 and 1. Then we choose a function that propagates the set members out from the origin. At whatever stage of propagation we choose, we determine what proportion of our original points are in what regions of the manifold. That is our "probability distribution."

This was just an intuition, and of course it lacks formal rigor. But as soon as I "saw" this, I realized, "Oh! so of course we could have, say, a normal distribution mapped not just on a plane, but also on a torus, or a sphere, or a Klein bottle."

I previously had never heard of such distributions, but after "seeing" this concept, I looked them up and, indeed, one can.*

So, Shonk, Nathan, others, I am sure that I have expressed this clumsily, but I also assume that a more rigorous definition that captures my intuition exists. Where is it? What is it called?

Furthermore, a sense I connection here to braid theory. Is that correct?

UPDATE: My friend queried his officemates who actually work on statistics on manifolds. They told him my definition is essentially correct.

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* It is very amusing to me that what is discussed in the link above is a type of "von Mises" distribution: in the last trading company I worked at, my boss was a PhD mathematician who had written a book on Richard von Mises theory of probability, and while I was there I published a book on his brother's theory of economics. I told him that I could just about guarantee we were the only IT department in the world in which members had written books about both of the von Mises brothers.