Why does a voltmeter read lower across a load than across a supply?
There is one thing that has been missed out from your 'classic resistor example' - the internal resistance of the battery.



If you simply measure the battery voltage with a voltmeter you get a higher reading due to the fact that there is no (or very little) voltage dropped across the internal battery resistance.

The voltage measured across a load resistor (or bulb) OR the terminals of the battery will be smaller by an amount equal to  I * R(int). 

The connecting wire has very little resistance and so does not contribute to a significant voltage drop between the terminals of the battery and the resistor so the voltages at these two places are, in practice, the same.