"I need to get my point in"
So far the examples have been friendly ones. Drafting is at its most intense when you disagree.
Your partner is explaining why they think you should both go to their parents for Christmas. By the second sentence you've identified the flaw, which is that you went there last year. From then on you're composing. You're choosing the phrasing, anticipating their counter-argument and preparing your answer to that. You're waiting for them to stop so that you can begin. Whatever they say after the second sentence you hear only as delay.
Meetings are the same. A colleague is presenting a proposal, and you've spotted a problem with the costs. You spend the next four minutes polishing your objection and watching for a chance to make it. You miss the slide on which they dealt with the costs.
The belief behind this is that the conversation is a contest and your point is your move. If you don't have it ready, you'll miss your chance or make it badly. There's something in that. In an argument it's unpleasant to be caught flat-footed.
But look at what you end up answering. You composed your rebuttal to their second sentence, and they then spoke for another two minutes. In those two minutes they may have qualified what they said or conceded part of your point in advance. They may have explained that the reason is their father's health and not whose turn it is. Your rebuttal, delivered on schedule, answers an argument they're no longer making. They can hear that, and it produces the particular exasperation of "That's not what I said." The rest of the argument is then about what was said, and the actual question gets left behind.
The effect on them is predictable. A person who feels unheard in a disagreement doesn't move on to considering your view. They repeat theirs, louder or at greater length, in the hope of getting it across this time. You then hear the repetition as stubbornness. Many long arguments are two people each waiting for the other to stop so they can say again what the other didn't take in the first time. Each is sure that the other one never listens.
If you want to be persuasive, the listening has to come first, for practical reasons as well as polite ones. People are far more willing to hear your view once they're satisfied that you've understood theirs. The quickest route to being heard is usually to show that you've heard. "So the main thing for you is that your dad's not been well and this might be one of the last Christmases he's up to hosting. Have I got that right?" It costs you nothing. Understanding someone's position doesn't commit you to agreeing with it. Once they've said "Yes, exactly", they've been heard, and they can stop repeating themselves. For the first time in the conversation they're free to listen to you.
You also argue better, because now you know what you're arguing with. Your point about last year's Christmas may be irrelevant, or it may need putting differently. It may also turn out that you don't disagree after all. None of that could be discovered while you were busy composing.
The joke belongs here as well. If you're quick, a funny remark will often occur to you while someone's still talking, and the urge to deliver it while it's fresh is strong. A good joke is a gift to a conversation, and I wouldn't want you to stop making them. But a joke has the same cost as a rebuttal while you're holding it, and it has a further cost when it lands in the wrong place. If someone was working up to something serious, the laugh changes the mood and closes the opening. They laugh too, because it was funny, and they go home without having said what they'd meant to. If you're known as the witty one, it's worth asking whether people also bring you their troubles. They may have found that with you things get turned into jokes.
As for the fear of losing your point, write it down if you're in a meeting. One word on a pad is enough to hold it. Across a kitchen table, trust that a point worth making will still be there in ninety seconds. If it's gone by then, you can probably do without it.