I've always thought it was like this, maybe I'm wrong:

O(some constant) -- "nearby" that constant (maybe "order of magnitude" or whatever is contextually convenient)

O(some parameter) -- denotes the asymptotic behavior of some parametrized process

O(some variable representing a small number) -- denotes the negligible part of something that you're deciding you don't have to care about--error terms with exponent larger than 2 for example

Those last two notations are, formally, the same. To call a part negligible, we say it's asymptotically bounded above by a constant multiple of this expression, which obviously goes away as we approach the limit. The first one is a colloquial alternative definition that would probably be considered "wrong" in formal writing.