if your landlord pays for cold water only, is the hot water a service of the gas or electric company?

Is the hot water heater gas or electric? What he is saying is that he does not charge you a water bill. He either has his own well or he is going to pay the water bill for city water to your apartment. The electricity or the gas that is consumed to heat whatever water you use will be on your electric or gas bill that you receive.

Technically the landlord still pays for "all the water". But depending on whether your "hot water heater" is gas or electric YOU are the one who pays to "HEAT the water you use".

You don't pay for "hot water". You pay for the gas or electric that runs your water heater.

I pay for water, but essentially you are correct, you pay either the gas or electricity costs to heat it.