Certainly a second order effect. Makes the cut a tiny bit bigger than what would be expected from measuring the tool bit projection.
Relatively easy to calculate the magnitude of over cut for any given run out. Unless the chuck is very poor odds are that it will be un-important compared to the other frustrations involved in trying to bore exactly to size with such bars.
Using the chuck rather than centres means you cannot remove and replace the boring bar in exactly the same place without special precautions. Worst point when using between centres type bars is that its pretty much impossible to easily make decently accurate measurements of the bore without removing the bar.
General advice is to get the bore as close as maybe then turn the mating part to size.
Clive.