My short (and overly legalistic) definition would be attractive public activity with a persistent identifier. By attractive, I mean designed to attract attention, however little. Nobody posts on HN in order to have a record of their private thoughts or avoid public engagement, for example; people post their thoughts specifically in order to share them. Insofar as one engages in acts designed to attract attention (not necessarily approval), and maintains a persistent identity such that one's acts are temporally/spatially connected, then one is a public figure in proportion to the attention those acts attract.
The persistence is important. Consider 4chan, where people can post anonymously. The downside of this, especially to the casual reader, is that people abuse the anonymity to say hateful things - although most of the time it's kids trying out the language of sexism or racism to see how the words feel in their mouths, so to speak, in order to reflect and understand the wider society they live in. (This might sound optimistic, but my view is that the bigotry there is only skin-deep and serves a function similar to that of hazing.) The upside is that one can engage in experimental social interactions without consequences - for example, making an embarrassing social admission in different fashions to gauge the variety of reactions. Now, someone might make brilliantly insightful contributions in an anonymous environment, but if they have the same discussion a week later they must rebuild any intellectual or social consensus from scratch. With a persistent identity, one can trade to some extent on one's social standing. Some online fora, such as HN, make this standing semi-explicit through karma scores and the like. If you say something that I find counter-intuitive, your high karma (expressing others' approval of your previous contributions) signals to me that your odd-seeming remark is probably not the product of mere foolishness, but rather has some rhetorical or specialist basis.
One of the tricky things about the world we live in, where video and text live on and on long after verbal conversations are forgotten, is that private individuals are increasingly creating public records. We wouldn't say that Joe Blow was a public figure just because he went to his neighborhood bar once a week and held forth on the state of the world to his neighbors. Sure, he is in public, but his 'public' consists of only a small group. Unless he starts speaking of controversial matters like having a plan to shoot the mayor, those conversations are ephemeral and weighed by society as such - unlike the situation faced by those who make foolish comments on Facebook, say.
There was a man. He constructed a persona. That persona was public. The man was private. The man worked very hard to separate the two.
Here's the problem: we have no way of knowing what someone's motivation for privacy is, and such motivations can be good or bad. Privacy is a basic right (implicit or explicit depending on your legal system), but to the extent that a person acts publicly that right is necessarily attenuated. Most public figures maintain a single identity and seek privacy for certain activities (family life, financial affairs and so on). Where someone maintains a private identity we tend to ask why, because the ability to recognize and distinguish between individuals is a basis of survival and social credit. A famous musician or artist may adopt a pseudonym to develop a personal brand: 'Picasso' or 'David Bowie' are such unusual names that we want to know more about the persons who adopted them. On the other hand, a famous villain may adopt a pseudonym to intimidate enemies or awe the public. Or an innocent person with enemies may adopt one to hide from them while still exercising the right to public speech...there are as many reasons to obscure one's identity as there are people in the world.
_why obviously fits within the former case - but that's only obvious if you're within the community of people to whom he largely addressed himself, and familiar with its norms. to an outsider, such as the Slate author), the sharp divide between public and private identities is mysterious, enigmatic; few people go to such lengths to separate their identities in this fashion, and enough of those who do are motivated by questionable reasons that it's worth asking why in this case, at least for a journalist. This would not have happened if, for example, _why had published a message saying 'Oh hi, my name is Jonathan Gillette aka _why, and although Ruby hacking is totally awesome I need to take a break from this sort of work to focus on family/ personal/ whatever matters. Thus, I won't be maintaining Hackety Hack (etc.) for the foreseeable future and you won't be seeing me at Rubyconf or other public events either. Thanks a lot and happy hacking.' If one disappears suddenly like Judge Crater [1], then it's inevitably going to attract a lot of attention and invite speculation - and indeed it did. Insofar as one chooses to pop a balloon rather than simply let the air out of it, one can hardly be surprised at the echoes that result. Even now, _why opts to maintain an air of mystery, as opposed to spending a few minutes on the phone explaining 'yeah, I was just tired of the whole thing and was drinking way too much coffee, so I chose to retire and now teach high school/ breed gophers/ live as a hermit.' This is a perfectly reasonable choice, but so is the choice of the journalist to follow up a mystery that piqued her curiosity, and that of a great many others.
1. http://en.wikipedia.org/wiki/Joseph_Force_Crater