300SDL, engine turns over for 1-2 sec, then 'thud' sound and stop dead

[ah-kay]

Need some ideas to tackle this problem.

It happens a few times, normally in the most inopportune times. Turn the key, engine turns over for 1-2 sec then stop dead 'in-flight', nothing! I can get it started by doing the following:

1) Turning the key again. It will be OK 2nd or 3rd try.
2) One time my wife got stuck and it would not start with a number of tries. Jump the battery and still the same. When I arrived, it started at the first try ?? Car is afraid of me?
3) Wouldn't start this morning. Try it a few times and no luck. Put the car in drive and roll it backwards a little bit. Cannot roll it forward, curb is in the way. It started right away. It may or may be related with rolling the car.
4) Battery swapped from another car, same problem. So battery is ruled out.

Wife does not want to drive the car as it is doggy. What could be the problem? My plan is to take the starter out and take a look to see anything obvious. What shall I be looking for? Will replace it in any event.