What a complicated question.
I would say I'm generally more attracted to shy types, but since I'm much more likely to be engaged by a non-shy type (for obvious reasons), my friends and girlfriends tend to be middle/outgoing. My most successful friendships have been with outgoing types that are also capable of more subdued, 1-on-1 interaction. Serious relationships are too complex for me to cleanly break down into categories. I think I'd prefer a shy type for many reasons, but a middle/outgoing type would probably be healthier for me, though probably more stressful.
Like others, I don't tend to make the first move in most social situations, but in places where I'm used to being in charge - soccer (I've coached for years), and at work (I'm not the boss, but I do have much more freedom to make choices than most) - I tend to act more outgoing. This is partly out of necessity, but there's also a comfort aspect to it. It becomes a habit after a while.
I'm very shy when I have to interact with someone I like, I'm extroverted towards friends. I'm somewhat shy/middle with strangers.
I'm definitely, on average, more attracted to introverts. That, however, may be due to the fact that most extroverts strike me as more obnoxious than most introverts.
I think my friends are just about equally split on the extroverted/introverted orientation, though.
Have to agree with these two.
On related note, shy and confidence-lacking people are drawn towards me for some reason. I think it has something to do with me being an introverted invidualist (for example, I never try to blend in any social groups or circles, work on projects on my own etc). I guess they think I'm in the same situation as they are and try to find peer support. They are usually recognized by their habit of apologizing everything, explaing their every act and retorts like "guess I was an idiot again, huh?". It's pitiful, but I can't really be rude to them either.
I missed this on my first read, but I just saw it and feel compelled to respond. I have developed a habit of apologizing, etc, just like you describe and it's not at all due to confidence (not saying this applies to everybody, but surely I'm not the only one...).
When I was young (until college or so, when I was really allowed to make my own decisions), I excelled in pretty much everything I did, with what looked to others to be little or no effort. Because of that (or at least that's what I believed at the time), some people despised me. I found that a self-deprecating sense of humor and an apology here and there, allowed me to have less hostile interactions with these types without compromising my performance in school, soccer, etc.
Looking back, I stand by my previous analysis, but only to a point. It's clear now that my issues were definitely also caused by my seemingly aloof demeanor. I don't come across so aloof now, at least when I need/want to interact with someone, but those are tough habits to break after all these years.
Anyway, my point is/was that it could easily be because they don't have great people skills and are trying to get along with others the only way they've found that works.