John, seemingly, you got it right (judging from the way you formulated the question).
It is mostly understood as ones first name. (keeping in mind that Westerners would often have an arbitrary amount of middle names... it also may mean Any one of them, or all of them + last name, or all of them without the last name, or some stable or hyphenated combination of them)
You may say "зарезервировал номер на имя" + anything. Because "зарезервировал номер на имя" is a very common expression. But:
1) the listener will expect "first name" or "first name +surname" and will be a bit surprised.
2) the listener may find (or not) your way to say it incorrect. But understandable. It wouldn't sound 'non-native'. Rather native, but not logical/common:)
You also may say "зарегистрировал на имя" + nickname/pseudonym. It is ok.
"Greek origings of your name" would be universally understood as "of John".
I'm not sure if I met 'name' to mean 'family name' in history books. Haven't been reading them for a long time. If this happens, it is uncommon.
But: It is one of those cases when the difference can be interpreted not as a matter of 'correct/incorrect translation', but as a matter of culture/tradition instead. Like 'name' and 'имя' refer to the same.
If I was reading a book where 'name' is translated as 'имя' in the context of family names... i wouldn't blame the translator for a suposed blunder. I would interpret it as a deliberate choise. I could make such a choise myself.