Having seen your start diagram it is clear that the existing start switch disconnects the feed to the ON position when you turn it to start. So while your placement of the white wire appears to be correct, to replicate the exact conditions of key starting you will need to leave the key in the off position when starting as the other circuits within the engine management must be powered from a different source. So I would say the procedure would be:
1 Turn the key to ON position to bring up the usual dash indicators
2 Press the start button while simultaneously switching OFF the key
3 As you release the button turn the key to ON.
I am not convinced that that will work easily so a better solution would be to use a changeover button with the common connected to the power, the normally closed side to the ignition key feed replacing its existing feed, and the normally open side to the white wire to the starter module. I'll put up a diagram based on yours in a bit!!
In order to check the white wire on the button could you post a larger pic of the bottom left hand image in the set of 8 images of the starter button?