thanks so much for the help
It's not really an assumption. Atoms are neutral particles. An atom always has the same number of electrons as it has protons in its nucleus. Particles that are not neutral (i.e. that have a net electric charge) are known as ions.
Again, not an assumption. That's what a hydrogen atom is.
This is simply not true. What you were probably remembering was that Hydrogen has a valence of 1, meaning that it can form one chemical bond with another element. It's easy for Hydrogen to either lose its electron or to "share" it with another atom, thereby participating in a chemical bond.
If, by what hydrogen is "usually" like, you mean, "what state it exists in in nature", the answer is that it can exist in many different states, and we use different chemical symbols to explicitly indicate what we are talking about.
On Earth, if there are large quantities of pure hydrogen, it will exist in gaseous molecular form, i.e. in the form of two hydrogen atoms covalently bonded together. We use the symbol H2 for this molecule.
When we are talking about neutral atomic hydrogen, we just use the symbol H. It doesn't exist in this form on Earth, but in interstellar space, densities are low enough and/or conditions for molecule formation are unfavourable enough for hydrogen to exist in a gaseous neutral atomic form (single atoms floating around).
In really energetic regions heated up by young stars, hydrogen can also exist in ionized form (as a plasma...a gas in which the electrons have been stripped away from their nuclei and you just have this soup of charged particles). Chemists use the symbol H+ to refer to this form of ionized hydrogen. These gas clouds (nebulae) actually glow (emit light) because of the gas being in an excited energy state.
However, in this context (in astronomy), a very strange alternative notation is usually used. Instead of just H, the symbol HI (I being the roman numeral) is used to refer to neutral hydrogen and the symbol HII is used to refer to ionized hydrogen. In general for any chemical element, the 'I' means 'not ionized at all', the 'II' means 'singly ionized', and 'III' means 'doubly ionized' and so on and so forth. I realize that this is very strange. I think this notation originally comes from spectroscopy or something.
EDIT: I should also point out that, although in my previous paragraph, the world 'ionized' exclusively meant, "having an electron missing", in general, it is possible for ions to form through an atom gaining an electron. I do seem to recall vaguely from chemistry that there are some circumstances in which you can get H- (two electrons), but that is not what hydrogen "likes" to do (i.e. losing an electron is what is most favourable from an energy standpoint, not gaining one).
So that was too much info. The short answer to your question is that you are right that any particle that has exactly 1 proton in its nucleus can rightly be called 'hydrogen', but it cannot rightly be called a hydrogen 'atom' unless the electron is also present.
Separate names with a comma.