I completely empathize with the feeling that there can be a large gap between what you're capable of expressing (e.g. what's on your mind) vs. what people perceive from you if you speak tentatively or without confidence. I'm honestly not sure what has helped me most through the years. I still stutter, and for the longest time (all the way through my early twenties) this stopped me from expressing myself in many situations. For me, one of the biggest changes was deciding that I'd speak even if I stuttered, and just let people deal with it. In fact, at times I'd first tell people that I stutter, which removed a huge part of the anxiety for me (and consequently led to less stuttering too, which was an unintended but appreciated side effect). I think sometimes telling people that you're a bit nervous about speaking up might lead to a similar effect, especially if you are doing a larger presentation. But that's just speculation on my end; everyone's experience will be different around this.

At the highest level of abstraction, though I'm not sure what helped me through the years, confidence in a way begets confidence. So it builds on itself. This is why I recommend people seek opportunities to speak more, and to force themselves to try difficult things when it comes to speaking. It snowballs after a while, and you may find that your similarly increases. The whole dynamic of the difficulty of responding real-time is also something that I experience; it's especially true that introverts have real difficulty when put on the spot and asked to make a dynamic decision, or to speak before they're ready. My interactions with others have improved over time as I've practiced this; some of it, for me at least, is to be willing to speak even before your mind is made up, when the circumstances require it, but qualifying what you say appropriately. So, for instance, I'll sometimes tell people, "Having only thought about this for 10 seconds, my initial reaction is [blah blah]. But I might feel differently after reflecting on this later." In this way, I'm able to participate when the need arises without feeling like I've overcommitted to a perspective that I might not hold long-term.

Don't let this stop you from reaching your goals, whether it's to manage teams or to do anything else that requires a lot of communication. You can definitely do it — many, many introverts do. (Have you, btw, read Quiet, by Susan Cain? If not, I'd recommend doing so; it's very good when it comes to encouraging introverts not to get stopped by these things, and giving some practical advice as to how to advance). Practice makes this easier over time; and also, in the long term, competence does show through. If you'd like to practice in a focused way, one avenue is to join your local Toastmasters group. Many engineers who've worked with me have joined such groups and benefited a bunch from a safe environment away from work where they can practice these skills.

I wish you all the best. You'll do great! If you ever have any questions in the future, just reach out.