There is no "supposed to be".  If you have to open an account in your own name, then it is NOT included with the rent.  It is rare that it would be.

"supposed to be" no, some rentals included it and others make you put it in your name and pay for it. You have to put it in your name because you are the one who will be using the water and responsible for the bill.

Beets me because that you mean the bill would come to you.

Do not put the water in your name if it is going to be paid for by the LL because it should be in the LLs name.  What is in the lease?  If water is included in the rent then there is no reason why you would be putting the water in your name.  I've been renting since 1964 and have never had to pay the water bill; it has always been included in my rent.  I have never been asked by any LL to put it into my name.  I have a feeling your LL is trying to pull the wool over your eyes.

That might be a rule of the water company.

If it is included in the rent it should be in your landlord's name.

Ask your landlord.

That sounds like a great question to ask your landlord.  The most reasonable explanation would be water not being included in the rent after all.  

That's a good question!

Read your lease to verify that the water is included in your rent. If it is, talk to your landlord & ask him why you are required to put the utility into your name.