I READ THAT IF YOU ADD HYDROGEN TO WATER, THEN IT MAKES IT HYDROGEN PEROXIDE, IS THAT TRUE?

Water has the chemical formula H2O, and hydrogen peroxide has the chemical formula H2O2, which by comparison contains extra oxygen, not hydrogen. So it does not, indeed it cannot, form hydrogen peroxide. The fact is, hydrogen gas does not bond to or react with the water molecules, it just dissolves into the water. It does not create some novel molecule like H4O, which would be chemically impossible to form anyways. Therefore, hydrogen water and hydrogen peroxide are completely different things.