I'm not entirely sure, but i'll try:
Theory 1 = Atomic number is the same as the proton number. The amount of number a proton has determines the element that atom is. So, for example an atom has 8 as it's atomic number and another has 8, also, as the atomic number. Meaning they are both the same element. So, the same atomic number = the same element.
Theory 2 = Ions - because the thing that determine the type of element is the proton number (which doesn't change). However, the mass number of any element can change because of ionic bonding. So, in conclusion, two atoms with the same atomic numbers but different mass numbers is called Ions.
Sorry, if this turns out to be wrong... I'm not sure which one is correct or even if any of my theories is correct, but I've got my eyes set on the 2nd theory, it makes more sense... Again, sorry, hope this helped... :)