The first seems compelling to me, less mass means less over all energy to freeze.
The second less so. Hot water has a bigger surface area, which makes heat loss fast at first. When it gets to cold water's temp it should have the same surface area as the cold water making the advantage disappear.
That quoted sentence says that hot water loses volume compared to cold water.
The person you’re replying to is saying that hot water simply takes up more space as well.