Yeah, I don't think of shyness as a characteristic either. It describes someone's behavior to me as well. I also would never associate it with introversion since introversion/extroversion is how an individual takes in information, it's not necessarily their behavior.
I generally use the term 'reserved' to explain a 'shy' adult. I usually think of 'shy' as a term used to describe a child's behavior, not an adult. Therefore, I think of reserved as a term to explain the behavior of the adult. Whatever, semantics shemantics!
As for socially anxious, it is completely different than shyness. It's classified as a disorder (particularly social anxiety) since a person experiences discomfort or a fear in social interaction, which entails a fear of being judged by others.
Like honestly, I'm a pretty reserved person. I rarely talk to strangers while I'm alone but I have no problems talking to someone if I am approached or I need something. And yes, I am an extrovert. I can't even stand the idea of small talk. There is no such thing as small talk to me. I feel like every conservation I have should serve a purpose, not necessarily be frivolous to pass the time. I'm just not that talkative unless I feel comfortable in the setting. So I wouldn't say I am shy, since there is nothing timid about me, even face-to-face... but I do pick and choose who I wanna converse with.