OK. The difference is that your relay unit actually houses two relays on completely independant circuits. The starter cutoff relay is in there, as is the fuel pump relay. The only thing they share is the 12 volt supply for the coils. The starter cutoff relay circuit itself is identical to my '03.
Also, I missed one connection yesterday which renders my jumper setup incorrect, and I missed the fact the the ECU applies power to the lines on the clutch switch, it does not simply read them for grounds. That was stupid of me, just reading the diagram instead of getting the voltmeter out.
So, some corrections for which wires to test on the connector. I "borrowed" your pictures.
The second pic is not so important for testing since we know the starter works, it's just informative, I guess. The first picture labels what should be happening at the wires. The one labelled "Connects neutral switch to neutral light" is actually dead when the connector is unplugged. Unplug the connector and use an ohmmeter to test the conditions I labelled. You should see that the neutral switch wire grounds when the bike is in neutral, and you should see the sidestand switch wire ground when the sidestand switch wires are connected. (That wire is the one you should just cut and ground directly.) Then connect the ohmmeter across the two clutch wires, and you should see the clutch switch work with the pedal: it's normally open, it closes when the clutch is operated. It's possible, if the pinout is different from '05 to '06, that I've got one of the clutch wires wrong on my labels, but I hope not.
Now connect the relay, switch your meter to volts on a scale good for 12 volts. Turn the ignition on, do not start. Ground the black lead of your voltmeter, and at the clutch switch wires, look for these conditions:
One wire (I think blue/yellow) will show 12 volts if the bike is in gear
and the sidestand is down (wires disconnected.) The other one (black/yellow) show 12 volts when the bike's in gear. This is what's controlled by the network of diodes in the relay assembly.
If that all tests out correctly, then listen for the relay clicks again: kill switch on, ignition on, bike not running, sidestand switch wires not connected (or wire at the connector not grounded.) Moving the shifter between first and neutral should click the relay. If so, move on. Put the bike in first. Operate the clutch. You should
not get any clicking. Now connect the sidestand switch wires, or just ground the wire from the relay connector, and operate the clutch again. Now it should click with the clutch switch closed.
I sure wish someone near you with an '04 or '05 could ride by and lend you the relay box long enough to see if it acts different with their relay. That would let you know for sure if it's the relay or your wiring. I think those circuits are actually the same in Gen-II bikes, but the relay part number is different. The '04 and '05 both use the part number in your picture.
BTW, you can tap any switched 12-volt wire and connect it to the brown/red wire at the meter connector to enable the neutral light.