Steve, you are spot on with those circuits, yes that is what they do, however your analysis is missing something.

A capacitor and an inductor both store energy. So let's suppose you had a charged capacitor for a moment.

What would be the quickest way to discharge that capacitor, with a 100K resistor or a 1K resistor? I think you would agree that a 1K resistor would discharge it quickest would you not?

The voltage on the capacitor would discharge in an exponential curve who's characteristics are that the most rapid rate of change occurs at the start and the rate of change in voltage slows down as the capacitor becomes increasingly discharged. Lets say the capacitor's initial charge was 40V but you are interested in making the transition from 5V to 1V as quick as possible and you were not worried about the time it took to reach 5V. So one strategy would be to discharge the capacitor until it reached 5V with a resistor and then short out that resistor. Of course in the real world there is no such thing as a short only varying degrees of small resistor. So let's modify that by saying discharge with a 10K resistor and when 5V has been reached add an extra 1 ohm resistor. The newly introduced resistor would introduce a new exponential curve in the rate of change in voltage resulting in a more rapid transition between the two voltages you are interested in.

Now a capacitor and inductor are very similar in that they both store energy and in both cases the way to remove that energy is to put as bigger load on it. However with a relay you have a pull in current and a smaller hold current. This corresponds to two strengths of magnetic field. What you want to do to make the contacts break quickly is to minimise the time it takes to go from the hold magnetic field to zero. So one way to do this, just like the capacitor, is to drain the energy ( magnetic field ) from the inductor slowly so that it stays above the holding field for as long as possible. Then when the field drops below the holding field their is very little energy left in it and will quickly dissipate.

So by introducing a resistor in series with the diode you are reducing the discharge rate of the inductor's field and making it hold up longer so that when it drops their is little energy left to keep the field close to the holding field thus making the contacts open more quickly. You care little about the overall discharge time because a relay is slow anyway. So you delay or make longer the discharge time of the inductor's energy so that the drop off at the end is quicker.