I'd guess the example of the glass of water was made as an afterthought, or as a vivid example of the potential complexity of the theorem. As your example shows, it simply cannot be true. If you think of a vase with marbles, it is also clear. Or if you think of an (ideal) gas in a closed system: would there be at least one particle that always(!) stays in the same place?

Show me a mathematical continuum in the real-world. And this is just the domain, nothing said about the mapping yet (the continuous function). Every analogy has its limits.

The space represented in a flat piece of paper.

The space is continuous even if we can only measure down to the Planck length.

Key here: "represented"

I'm only a mathematician, no physicist. But I think to remember, that the concept of a continuous physical space becomes quite muddled at this scale.

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If you get to assume continuous paper, then I get assumed continuous paper.

If you don't want to assume continuity, then you get it back by rephrasing your theorems as "within a margin of error equal to the distance between discrete objects".