[1] http://www.yourcoach.be/blog/wp-content/uploads/2014/02/Dale...
I highly recommend. It gives you a powerful alternative and a beacon that you follow every time you have a conversation with someone. Although the book is old ( before the Internet ), it has pretty much summed up everything about people's nature and can teach you what your parents / teachers forgot.
1. Say less.
2. Listen more.
3. Think of everything from the other people's point of view and honestly consider what is best for them.
I think one thing you said was quite telling: "you go into a meeting with your mind made up". I'd suggest trying to being open to having your mind changed is a good first step too ;)
No one knows what you believe
The few people I've met like this were pretty much office bullies. There's no mythical silver tongued nice guy. When someone dominates the meeting and shoves his agenda into your face and argues until you're exhausted, well, that works because he's not worth fighting. The question is do you want to be a person like that? Pretty much anyone can be taught to be a Gordon Gekko or a GlenGarry Glen Ross's Blake.
I don't take my job remotely seriously enough or care to modify how I act for crap like this. I kinda pity those who do. They seem to get their validation and emotional fulfillment only from work.
The above might be overly critical, but I think everyone hits a point in their career where they have to decide where they will draw the line in terms of personal modification to get ahead. There are so many more asshole-ish ways I could be at work that would definitely lead me to a higher level of success. I just consider it against my moral system to be a person like that and think its absurd that we have to do this constant study of "how to get ahead" just to make a decent living. Threads like this just depress me. Work life shouldn't make this kind of demand on our personal and internal lives. I think if everyone pushed back more against employers, we'd all be happier, instead we have this rat-race that we accept as perfectly normal and rational. Its not.
Anyway, that's the kind of thing i'm talking about. It's easy to talk over people and beat them over the head til they do what you want. I'm more interested in getting everyone on my side and excited about it.
Persuasion is independent of "shyness" (actually, being an introvert) but a lot of it has to do with natural talent, unfortunately [1]. You can definitely improve it, but some people are just born with a higher degree of "energy transitivity". For instance, their face have more muscles and they're wired in such a way that it is easier for them to control it, among other things.
[1] Malcom Gladwell wrote about this briefly in "The Tipping Point". You can learn more on this by looking up the book "Emotional Contagion" by Elaine Hatfield, John Cacioppo and Richard Rapson. There's also Howard Friedman, from the University of California, who created the Affective Communication Test, which aims to give an individual a measure of his personal carisma - this transmissive or "magical" ability you speak of: http://www.faculty.ucr.edu/~friedman/act.html
The Dale Carnegie Company also offers instructor-led training in small classes. Having taken it, I would recommend taking the in-person course if you can; nearly every major city has a DC office. If you're willing it to take it seriously and put work into it, it can be quite beneficial.
I would also say it's kind of magical.. some sort of energy transmission..
The way you talk about it reminds me of people's account of Jobs' so called "reality distortion field". However, it's not some mystical thing that some people are born with. It's a learned skill just like software engineering.
Many comments have pointed out "How to win friends and influence people" so I'll start there.