That diagram does show a four wire solenoid. The red battery cable, the black down to the starter, a blue/white from the start button, and a fourth that is off the bottom , a black wire, going to a set of triangular lines. These triangular lines represent ground.
You can see the same symbol in several places in the diagram, it always means ground.
As RG said power comes to the solenoid through the key switch, kill switch and safety relay. This power goes out to ground to complete the circuit. This trips the relay, sending full battery power through the large cables to the starter.
There are several tests you can do to test the starter. With a set of jumper cables like to jump a car, hook one end of the jumpers to a good battery. I have a spare car battery I use for these tests. You can even use your car battery, just get the car and bike close enough to reach from one to the other with the jumper cables.
Hook the cables to your battery, Hook the ground side to one of the front foot peg mount nuts, this is a good ground. Put a large screwdriver in the positive clamp.
Now from under the bike peel back the rubber cover over the big cable at the starter. Touch the screw driver to the cable lug on the starter. This may cause a few sparks but it should spin the starter and turn the engine. If it does spin the starter and turns the engine, you can be pretty sure the starter is ok. Now put the rubber cover back over the cable lug.
Now move up to the solenoid. Peel back the rubber covers from both cables. Touch the screwdriver to the cable that runs down to the starter. If the starter worked in test #1 It should work this time. If not the cable is suspect. Just cleaning the cable ends and lugs should improve things If not replace the cable.
Now if your bike battery is fully charged unhook the jumper cables from the spare battery. Use just the red side to jumper from the cable from the battery to the cable to the starter. This should spin the starter. If not it could be bad cable connections, bad cables or bad battery.
If all these tests prove the starter and bike battery are ok then unplug the blue/white wire at the solenoid. Hook a jumper wire from battery positive to this blue/white wire. This should spin the starter. If not check the ground connection on the black wire on the solenoid.
If the solenoid clicks but wont spin the starter then the solenoid is bad. A solenoid is just a switch. The relay part just closes the switch. If the switch contacts are burnt no power will flow from the battery to the starter.
If the solenoid spins the starter in this test then the wiring between the battery and solenoid needs work.
You say it kick starts and runs ok. Do all the lights and such work ok?
If so you need to check for battery voltage on both sides of the kill switch. With the key ON one wire should have battery voltage on it at all times, the other only when the button is pushed. If power wont come through the start button then it can be taken apart and the contacts cleaned.
If the start button has battery voltage on both sides when you push the button but the starter won't spin then it could be a bad blue/white wire from the start button to the solenoid. At the solenoid test for voltage on the blue/white wire when the start button is pushed. If no or low voltage the wire needs to be traced to find the bad spot and fixed.
Leo