A better example I've seen for the theorem is that if you take a paper map of a country, messily schrunch it up into a ball and drop it somewhere in the country, there will be at least one point on the map that is exactly above the corresponding real point in the country.
As others have said, it's meant for mappings of the space to itself. So stirring the water, but not moving the glass.
But anyway, the theorem works only for continiuous mappings. The moment they started mentioning "water particles", instead of some hypothetical fluid that is a continuous block, the theorem no longer applied. You could break it by mirroring the position of every particle. There's still a fixed point (the line of mirroring), but there's no obligation that there's a particle on that line.
As others have said, it's meant for mappings of the space to itself. So stirring the water, but not moving the glass.
But anyway, the theorem works only for continiuous mappings. The moment they started mentioning "water particles", instead of some hypothetical fluid that is a continuous block, the theorem no longer applied. You could break it by mirroring the position of every particle. There's still a fixed point (the line of mirroring), but there's no obligation that there's a particle on that line.