This is the cap with the spring that I was referring to (the red arrow is pointing to it)...
That's the socket, that's HELLA DIN plug. Should seat. Beyond that, I've got not much else...
Did you buy the bike new or used? Looks like the contacts in the socket have been spread apart. They (the contacts) are supposed to lip over the first part of the plug and seat / contact it. Do you have a pair of needle nose pliers, maybe bring the contacts a little closer so they mate with the copper part you see there (power). The outer of the HELLA DIN is ground and contacts to the outer wall of the socket. Should work for you.