Is Cold water heavier than hot water?
The "counter argument" you link to cites E = mc^2 and the higher energy content of hot water.

First off, note the "c^2" in that formula. That's the speed of light, squared. It is a huge number, meaning that you need huge differences in energy to get tiny changes of mass.

The other way around, you only need little mass to generate lots of energy, something nuclear physicists have been exploiting for quite some time. ;-)

So, while that effect is "true" to some extend, it is tiny. But yes, one molecule of H2O has more mass when hot than the same molecule when cold.

But when we talk about "weight", we usually mean "density", i.e. weight per volume. And here, a different, much more pronounced effect comes into play: Most substances expand when heated, reducing their density. Each molecule might be a little bit heavier, but the molecules are much farther apart, more than offsetting the E = mc^2 effect.

Cooling down increases the density -- the same amount of matter takes up less volume. An effect that home preservers have exploited for generations when canning things: You put hot produce into a container, and close it. The contents cool down and contract, producing underpressure in the container, sealing it. That's the "pop" you hear when opening a glass of marmelade the first time.

So, per molecule hot H2O is (a bit) heavier, but per volume hot H2O is (much) less dense.

At least down to about 4 degrees celsius, where a peculiar quirk to the H2O molecule makes it become less dense again with further cooling. That's why a pond starts freezing at the surface, not the bottom, where the 4 degree water is up to the point where the whole pond is frozen solid. An effect that fish have been exploiting for quite some time now. ;-)