Let me put it this way.
In the burette, you are measuring volume delivered. If the burette is not dry, the extra volume will reduce the concentration as you have noted, and therefore affect the totat volume required for the end point.
But in the conical flask, you know the volume you introduced into the flask right? For, eg. if I know I put in 50 ml of solution into a wet conical flask, no matter how much water I add to it, I still know the original volume was 50 ml. Any extra water will be excluded from the calculation. The titration will tell me total moles. The conc in flask is then tot moles/50 ml.