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Isn't a proton the hydrogen ion H+?

Technically, yes. But for convention, no.

Your question also applies to alpha particles. From: http://en.wikipedia.org/wiki/Alpha_particle

Some science authors may use doubly-ionized helium nuclei (He2+) and alpha particles as interchangeable terms. Thus, alpha particles may be loosely used as a term when referring to stellar helium nuclei reactions (for example the alpha processes), and even when they occur as components of cosmic rays. However, helium nuclei produced by particle accelerators (cyclotrons, synchrotrons, and the like) are less likely to be referred to as "alpha particles" because the high energies produced by these sources highlights the striking difference in behavior of their particles from the classical alpha particles produced (and originally defined by) the process of radioactive alpha decay.

The same goes for a proton beam.

The parent is correct, but to throw another shoe in the gears, there are other situations in which a proton is referred to as a hydrogen ion. For example, during reionization in cosmology, hydrogen becomes ionized. The H+ ion is then referred to as hydrogen. Sometimes in chemistry when dealing with acids, H+ is referred to as hydrogen and not a proton, but I've also heard it the other way.

I think a good guess, is that if you're dealing with nuclear physics, call it a proton, anything else call it whatever you want.

Atoms have neutral charge.

wikipedia://ion says:

An ion is an atom or molecule in which the total number of electrons is not equal to the total number of protons…

The H+ ion (i.e. proton) is an instance of atom under this definition, as are all ions.

And wikipedia://atom says "The atom is a basic unit of matter that consists of a dense, central nucleus surrounded by a cloud of negatively charged electrons." If you're going by quoted Wikipedia definitions then H+ isn't an atom, so therefore it isn't an ion.

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