I draw a joke distinction between a geek and a nerd like this. A geek is someone who loves technology and toys. A nerd is someone who gets utterly obsessed by a particular topic.

I usually bring this up when explaining to a fellow programmer that I'm a nerd but not a geek. Which fact I then demonstrate by pulling out my flip phone.

If they proceed to doubt that I am a nerd, I'll then offer to explain something esoteric. A good example is the differences between the Riemann integral, the Lebesgue integral, and the Generalized Riemann integral. And why if the last had been discovered earlier, we might not have needed to develop Lebesgue integration at all!

Certain kinds of nerds do (though usually not geeks).

Seems like pretty handy categorising - it's come in handy already! :P

Though not typical, it's also worthwhile to remove the "socially awkward" connotations of the use of "nerd" and "geek" terms and ascribe those to "dork" instead. Many self-described geeks and/or nerds don't consider themselves socially handicapped and it's deleterious to society to muddle academic engagement with social impairment.

If the programmers around me self-declared as nerds I would reverse the distinction. Because I don't care about the label at all. What I care about is concisely explaining why it is that you can find someone who is competent at programming who does not share a lot of the interests that they assume I should share.

I usually bring this up when explaining to a fellow programmer that I'm a nerd but not a geek.

I believe this answers your question. Being a self-proclaimed nerd, btilly clearly does, per his definition.

It seems to me they are too, well, mushy to be really useful.

Both words seem to be used to describe people that like specific categories of things but are unpopular or awkward. I refuse to be labeled with either regardless of any fine distinction in meaning. To me, what you are describing is like, "no I'm a douche, not an asshole." (I think you are neither, just an analogy.)

I have definitely left some in utter disbelief.

Among unpopular and awkward technical types, I stand out as being both awkward, and exceptionally extroverted. It is an interesting combination, though best observed from a distance. Kind of like a train wreck.

But you have to work with what you've got. And somehow it does seem to work out for me in real life.

We might as well be talking about purses.