The glow plugs have nothing to do with whether or not the starter engages. It sounds like the glow plug saga warrants its own thread to try to iron out what you have and the best way to go about straightening it up.

The starter definitely sounds weak in your video clip. You could hear when the bendix DID engage, the motor was too weak to spin the engine over. I'd be willing to suspect you have a poor engine ground or poor battery connection somewhere and that's where to start troubleshooting. If you have a poor ground (engine to chassis strap, or chassis to battery cable) its entirely possible for the motor to be functioning correctly, but due to the poor connection lacking enough current for both the solenoid to fully seat and the motor to have any power.

Barring a bad connection, the starter motor may be in need of service/rebuild.