Hello @Jens.Peter.Nielsen , You were right! I could assign the Start button of the MPK Mini Play to the Run button.
The Stop button cannot be used because you can only assign one button from the controller to one button into VCV. So, as you suggested it, I also changed the Transport parameter of the MPK Mini Plus to ON/OFF from its program editor and it worked. 
I would say that it’s not a very elegant solution but it’s quite simple to implement and it works. 
Many thanks for the help @Jens.Peter.Nielsen and for all those information you shared. I hope that this issue could also help other members of this VCV Rack community who have a MPK Mini Plus to control it.