You are being confused by linguistics; an attempt to explain, that has failed. Elements are defined by the number of protons in the nucleus. In the neutral atom there are an equal number of electrons in their appropriate energy levels [orbitals]. The electron arrangement is responsible for Chemical reactions and varies with the element. You will learn all about that in chemistry, college and graduate school so be patient.
The nucleus of hydrogen, Protium, H, has one  proton. There is also another stable isotope [nuclide] of hydrogen with one proton and one neutron, Deuterium [D]. There is another "natural" isotope of hydrogen, Tritium [T] that contains, again, one proton, and 2 neutrons This isotope [or nuclide] happens to be radioactive or unstable. Other isotopes of H have been made in nuclear reactors. This is true of most elements: there are several stable isotopes, sometimes natural radioactive isotopes, and many others have been made in nuclear reactors. The artificial isotopes are usually of little concern in most chemistry unless inserted specifically into molecules as tracers. In Chemistry we are usually talking about the natural mix of isotopes unless specifically stated or sometimes strongly implied.
Isotopes were formed in the nuclear reactors of stars, and we get to work with those that are stable enough to form into planets. An elemental symbol in chemistry almost always refers to the natural mix of isotopes the only ambiguous one is H. H can stand for the natural mix or for protium, the H isotope with one proton and No neutron. Isotopes of an element have very similar but NOT identical chemical and physical properties. This means that chemical reactions carried to completion or equilibrium will maintain the isotopic balance. Other reactions will show isotope effects. The differences are most pronounced for hydrogen H, D, and T.
Tritium, T, gives us a hint to what is going on in nuclei. It has an extra neutron and is unstable. Neutrons are needed to prevent the positive protons from repelling each other but too many neutrons are unstable in themselves. There are optimal neutron numbers for each element, usually close to the number of protons in each direction. Too many or too few neutrons and the nuclide is unstable, radioactive. It is fascinating to lookup a table of nuclides to see which are stable and the various nuclear reactions of unstable nuclei to become stable. A nuclide with too many neutrons loses negative charge; a nuclide with too few neutrons loses positive charge; there are several ways of doing each. This is nuclear chemistry!
The mass of a proton and a neutron are almost identical; the mass of an electron is much smaller so the mass of a proton+electron is again almost the same as that of a neutron. Each nuclide [isotope] consists of a number of proton-electron couples and a number of neutrons. These can be counted, and the count is the Atomic Mass Number. The total number is measured by comparing the mass of the nuclide to the mass of a defined nuclide and relating it to Carbon-12. The atomic mass unit, amu, is 1/12 the mass of a carbon-12 atom. [this is not strictly true because the definition is slightly different based on a defined value of Avogadro's number]. The number of proton-electrons and neutrons is measured by the relative mass; the number of protons is measured from Xray spectra; the difference is the number of neutrons. in the isotope. Tha actual relative mass in atomic mass units must be measured and is very close to a whole number because the mass of a proton-electron and neutron are almost the same and the binding energy holding the nucleus together, while huge, requires a small change in mass.
Isotopes exist and chemists usually use them as they are in the natural mix. There are specific applications where specific isotopes are involved. Chemists are not commonly involved in manufacturing isotopes. That is done by nuclear chemists and physicists using nuclear reactors, colliders etc. It is going on in the Sun and stars, a little bit in our upper atmosphere from cosmic ray interaction with matter, and the decay of radioactive nuclides that are still around, as we watch.