It certainly helps to troubleshoot the right way. The terminal you were touching with the battery lead was the solenoid terminal, not the main terminal to the starter motor. All you've proven is that the starter solenoid works. The battery + has to be connected to the main terminal on the solenoid for it to pull in the starter motor.

When you're turning your key, do you hear the same clunking you got from that video? If so, you either have a bad connection, a bad starter motor (unlikely), or the contacts in the solenoid have failed (replace the solenoid).

As to the buzzing noise, who knows? You were 15 feet away from the car when you made the video, it could be anything. Get under the hood and find out what's making the noise! It shouldn't be hard to find.

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Current stable:
1995 E320 157K (Nancy)
1983 500SL 125K (SLoL)

Gone but not forgotten:
1986 300SDL (RIP)
1991 350SD
1991 560SEL
1990 560SEL
1986 500SEL Euro (Rusted to nothing at 47K!)

Gone and wanting to forget:
1985 524TD 167K (TotalDumpster™) [Definitely NOT a Benz]