Damocles, If you take apart the high ampere switch in the middle of the positive battery cables on its path to the starter motor, you will find a "plunger" inside of it. It is large enough to make a pretty distinct "Click", when energized by the smaller low ampere switch which is energized by the button.
The plunger on an authentic starter solenoid mechanically engages the starter motor with the engine. Don't think the "plunger" inside the KLR650 starter relay performs that function; rather, I expect that "plunger" within the relay merely closes electrical contacts energizing the starter motor.
But . . . I've never disassembled a KLR650 starter relay.
The genuine starter solenoids also perform a relay function; however, they earn the name, "solenoid," from the electro-magnetically controlled plunger mechanically connecting the starter motor with the engine on starting.
Regardless, English is a fluid and changing tongue. As mentioned, "solenoid" is used equivalently with, "starter relay," nowadays, even though no electro-magnetically controlled plunger from a solenoid may exist to temporarily engage physically the starter motor and engine on starting.
And . . . one might say, ALL relays are "solenoids," because electromagnetic force is involved in their operation. Yet, if the STARTER RELAY is a SOLENOID, is not the STARTER CIRCUIT RELAY also a "solenoid?" As well as the FAN RELAY? What is the distinction between the starter ("solenoid") relay and the other relays?
The solenoid-actuated plungers connecting the engines and starter motors are clearly shown in the Internet images yielded from a "starter solenoid" search.