The idea that some people are more technically focused than others works for me as a definition of "geek". Especially I like when people use the word in a way that accepts that some people are simply like that, and that that's OK for them even if it's foreign to a majority of the populace.
Like you, I'd prefer to consider "geek" a category with a permeable boundary. I'd argue however that there are many people who spend practically their entire lives inside the category and others outside the category.
Donald Knuth seems admirably sensible about many things, one of which is publicity. I see him putting himself before the public where he is thereby able to accomplish something that he considers worthwhile, profitable. So I think that there's a side we don't see and that the side we see should surprise us if we imagined that geek behavior is whimsically self-indulgent. Nevertheless the word "geek" fits his activity that we see. He concentrates his attention as much as he can in a delineated technical arena where he seems unlikely to meet with conflict. Look at the good that he does and the grace with which he does it. If that's being a geek, it must be a really good thing.
Take a look at the (terrible, imo) TV show Big Bang Theory. The main characters are purposely socially awkward and distant.
Real-world geeks are typically far from that: far more exist on the socialable side of the spectrum than those who are constantly portrayed in the media. :(