I think the confusion is that some box scores list LOB just like at bats, runs and hits. In those totals if three players strike out with the bases loaded they are all counted as 3 left on base (for each batter) so when you add up the LOB's from the box score it will be a much higher total than the actual number of runners left on by the team. Team LOB totals are only those left on at the end of the inning but individuals LOB totals are calculated for every at bat.
As for the original question I don't know if moving the baserunner still counts as a left on base or not. It shouldn't but who knows.