“Actually, wait, that’s not exactly how it went” makes people trust every other story you tell.

Correcting yourself mid story costs a moment and buys a reputation. People quietly track whether your accounts hold up, and a small self correction is the strongest available evidence that you are not smoothing anything. It says accuracy matters to you more than the shape of the story does. Everything else you say inherits that credit.

When you notice you have drifted, say so right away instead of letting it stand. Do it lightly and keep going, since no confession is needed. Over time this is what makes people believe you when it counts.