So, yesterday I was at a day-long training
listening to a key-note speaker who talked about "reaching introverts." He said, "You know, the socially awkward who are all in their heads, not making eye-contact."
That offended me. I'm not "socially awkward."
Yes, I'm often "all in my head," so busy thinking about something that I'm not making eye contact or interacting with those around me. I don't consider work or chores at home a social environment, and I LIKE being "all in my head." I wish extroverts would realize that interrupting is still rude when you are interrupting internal thought rather than external conversation. I often wonder if they aren't "in their heads" because whatever is in there won't spark without someone from without poking at it.
When I'm in a social situation, by choice or not, I know how to interact with people. I'm not "awkward." Those who are uncomfortable, who are "awkward" with themselves are uncomfortable around those of us who are not, and so project their "awkwardness" on to us.
Yes, there are socially awkward people, both introverts and extroverts. But being an introvert does not automatically make one "awkward."
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I'm a solitary person but work in a field that requires me to present to groups of strangers, assisting them largely to learn how to be more effective educators.
Being effective means being a good listener more than a yakker.
Most people in a conversation are just waiting for their chance to yak in a one-directional manner.
Who does this person think he is to use a term like "socially awkward"?
As you suggest, extroverts can be, and often are, most awkward in a public or social setting.
They are quite unaware of how they come across to others, see my other post on this thread for an example.. 
are just waiting for their chance to yak in a one-directional manner."
That's my experience.
"Great minds discuss ideas; average minds discuss events; small minds discuss people." - Eleanor Roosevelt
A great many people are uncomfortable discussing ideas because they don't really have any, all they can talk about is people and if they really strain they can maybe talk about events.
It's difficult not to just veg out on many people, their conversation lacks any kind of spark of creativity.
I was (unwillingly) listening to one side of a phone conversation the other day, the person on the end I was physically present for is someone who almost never stops talking, their big complaint of the conversation was that people can't just listen, they have to interrupt. 
If you don't interrupt this person you'll never get a chance to say anything and they are horrible about interrupting anyone else who happens to be speaking, I feel like I'm standing under a waterfall of words pounding on my skull almost every time I talk to them. They also like to make eye contact and get really close, practically nose to nose and it triggers my flight or fight reflex unless I deliberately suppress it.
I don't want to make stupid small talk with people. I will be pleasant and like to make people laugh. I have social skills. I will make sympathetic noises to the checker at the store, and say something like "I bet your feet hurt" or "It's probably been a long day" and they appreciate that. I can be very entertaining.
The familiar part is where you say you are introverted unless you are around artists. I am an artist and a serious musician on two instruments and voice. I have performed in two plays in the Hispanic community and was a soprano in the chorus of an opera done by a community opera company. That was great fun and we did it in the original Italian, of course. I can have a great time talking about the latest exhibit at the art museum, or the fact that Houston Grand Opera is in the middle of a very avant-garde Ring Cycle now, having presented two of the operas, or many other kinds of music & musical artists. Or jazz chord progressions. I've seen enough of the standard operatic repertoire that the usual productions of the usual works bore me and I won't go see them.
Other good topics are philosophy or history or economics. I know a bit about those. I also have a few other topics I am "cracked" on and know a lot about, generally law and biology (have degrees in those) and Romance languages (took Latin, Spanish and Italian in school).
But your average TV show, or your average celebrity tabloid, I have no interest in. And of course no interest in conservative religion and conservative politics. If I admire a celebrity, it's because they've done something I consider worthwhile with their lives as an entertainer or an educator, or they've spoken out on a cause that shows that they are thoughtful. There are people who are famous for being famous, and people who are famous and talented at what they do. To quote Lewis Black, "WHY should I keep up with the Kardashians?" 
If I am around people interested in what I am interested, I am right there and listening and contributing. The average dolt who is into stupid TV shows and shitkick music and hating Obama, nope. I will get up and leave. Just about all my relatives are dead or estranged and so I don't have to get into any political arguments with them. My parents were yellow dog Democrats, though.
I think a lot of so called extroverts just want to dominate and control other people. By talking over them, being loud, telling them what they should do and so forth. They don't listen. I don't want to dominate anyone so I have to assert myself to not be dominated. Sometimes I am too nice and want to get along with everybody, and some people you just can't be nice to. You have to get away from them.
