You’d need on the order of hundred times the mass of Saturn to create a red dwarf, so the swimming pool might actually be enough to kick it into being a Sun-sized or even larger star.

Right, and Saturn is less dense than water because it is mostly composed of air. It's like a sponge with the spongy lattice removed. It's not a solid. (At least, the vast majority of it by volume is not a solid.) That's why it's called a "gas giant".