our bodies transfer heat from our body to water on the surface of our skin, water that then evaporates, resulting in heat loss from our body
Not a great explanation. Sweat is obviously body temperature, so heat doesn’t move into it. Unless it turns into water vapor, which is a process that requires quite a lot of heat energy, cooling the remaining liquid or the surface.
I don't see anything wrong with this explanation? Sweat (or more pendantically the water in sweat) does evaporate and that effect does remove total heat from the body.