If I'm looking at it correctly, the first picture is the connector for your right-hand switchgear. The problems you describe don't involve that area, so I'd just leave well enough alone with that particular connector.
The fact that it's only starting with the clutch in is a clue to the problem(s), but not a definitive one. The starter button gets power from either the instrument cluster when it senses neutral, or the clutch switch.
So, if the start button is not getting power in neutral, that means either the cluster is having trouble receiving the neutral signal from the gear position sensor on the back of the transmission, or it's getting the signal but the outbound wiring from the cluster to the start button is bad. Both are quite possible, but it's the cluster connector that seems more often at fault. Since you have the cluster off, you've already disconnected those two -- the two rows of pins on the back. So you want to clean those pins really well, and also do the best you can with the sockets that receive them on the wiring harness. There's not really much you can stick in the sockets to clean them with, but you can spray contact cleaner right in them, and then give them plenty of time to dry before re-connecting them.
(Also, if you mean it'll only start with the clutch in when you have the cluster removed, that's to be expected as there's no neutral signal to the start button when the cluster is removed.)
