The number of protons in the nucleus of an atom determines what element the atom is an atom of. If there's one proton in the nucleus of an atom, then the atom is an atom of Hydrogen. If there's any other number of protons other than one in the nucleus of an atom, then the atom is an atom of some other element. It's not an atom of Hydrogen.
If you say you have an atom of Hydrogen, then I know there's exactly one and only one proton in its nucleus. If you call it Hydrogen-2, I know you must be talking about the Atomic Weight of the nucleus, and there must be a neutron in there too.
If you called it Hydrogen-37, I would know that there would still be exactly one proton there, lost in a crowd of 36 neutrons.
