Possibly your Starter Clutch
After listening to your videos, the first one with the view of the instrument cluster, it sounds to me that you may have a problem with the starter clutch assembly. It sounded like the starter began to turn the engine over, then the starter motor went into free spin, like a high speed drill, then the ratcheting sound happens. the starter clutch is sort of like a ratchet that only engages when the motor is resisting the force of the starter motor.
The other video of under the seat, the sound appears to me to be from a relay clicking. there are a number of relays underneath the side covers. Using a stethoscope or a screwdriver pressed between the relay and your ear, determine which relay the click is coming from and then refer to the component diagram in the shop manual to identify the circuit involved.
This site use to have a link to a shop manual if you don't have one. A Climer Manual is OK, but the Yamaha shop manual that use to be on this site is much better.
if you can't find one, contact me and ill try to send via email.