No Richard,
This is not supposed to be correct. The one shown (1st pic) in our Tech manual section is correct, i.e. the snap tool roll made of a multilayer canvas or vinyl.
It is not so much a question what one can find in his/her car but what was originally delivered by MB. For instance, my car had no tools at all and no books. So, did the car come without it? Most likely not.
Again, the spare part list(s) together with the "really original" cars (e.g. one-owner cars) tell you what the truth is.
The correct tool roll for a W113 is the 112 580 02 xx. The others with thin monolayer canvas and laces comes from W108, W114, W107 or elsewhere - and those have different part numbers and were slightly cheaper when new.
Best,
Achim
(no car, only parts & info collector)