Race is a social construct, and a recent shoddy one that doesn't even hold up to other social constructs like gender or ethnicity. It was invented to justify slavery and genocide in the Americas, and then became the favorite tool of all imperialism.
As for whether you are a member of any given "race", there are two ways to look at it. One is self-identification, which is as easy as it sounds: you get to choose for yourself. The other is how others look at you, and this is where racism comes into play. Depending on whether they view you and traits you inhibit as belonging to one "race" or another, they may treat you more or less favourably.
The clincher is that that too is a matter of identification, an arbitrary process. If you don't look the way people expect, racism might work contrary to its expected association with ethnicity. The William White example above would be a case of this. Also, because external identification is done by individuals, definitions of "races" may vary from person to person, group to group. I might be misremembering, but I believe the Oscars were praised by Americans a couple of years ago for including more persons of color among its nominees. The "issue" being that one of these nominees were an actor of no more complicated ethnicity than being from Spain. To said Americans, he was hispanic, and thus a person of color. To the actor himself, and many Spanish like him, he was clearly not.
So to answer your question, Magmacube, um, I have no idea. It ought to vary depending on who you ask, so maybe if you're very curious how other people view you a big survey could be of use. Or you could just go for the self-identification route and be satisfied with that.