You're assuming an absence of solar hot water heaters - the world isn't homogenous.

eg. Our household conserves water (we track every litre we bring up to the tanks and filter) and we don't have a dishwasher - hot water is heated by the sun and run-off goes to the grey water tank for the garden.

No I'm not. I don't care how you heat your water. I'm saying you (should) heat it either way, and you use more of it when hand-washing so if you do pay to heat it, you pay more by handwashing.

And in your case, you use more water because you hand-wash, than you would if you used a dishwasher. You use slightly less energy because of the pump but also a lot more of your time, so... obviously how you relatively value those different things is up to you.