in your first video the step-up solenoid is powering and staying powered. That's likely where the "buzz of death" is coming from and the vibration is just being seen on the reset solenoid plunger.

when you hear the buzz, see where it's coming from but don't leave the game in that state for a long time - a solenoid coil will get very hot, let out some smoke, and eventually burn up.

the one thing ya gotta watch out for when the playfield is tipped up is whether a game has gravity operated stuff like spinners ... or in your case, the flags. You often need to mechanically wedge those devices so when the playfield is not down, the switches those devices control aren't held closed. Or stick paper between the switch contacts.

pedantically, a solenoid has a coil with the plunger rod that sucks into it when powered. A relay has a coil that pulls the armature plate down onto the coil top when powered, and the armature plate moves a lifter with switch blades stuck thru it.

when measuring the resistance of a coil, you must ensure the coil is isolated from the rest of the circuitry. If you just stick the meter probes on the two coil lugs, you may be measuring resistance thru a roundabout path and not thru the coil itself. If you're comfortable reading the schematics, you can figure out how to ensure the coil is isolated (make sure at least one switch in all possible paths is open).

If not good yet with the schematic, you could unsolder one of the coil wires then measure.

however, most people just briefly touch a jumper wire from the correct power wire to the correct coil lug and see if the coil fires. The trick is learning where to attach the jumper wire ends so you don't blow a fuse.

alternatively, you can measure the voltage with probes on the two coil lugs when the coil is supposed to be powered and see if it's right. Again, the trick is ensuring the circuits powering the coil are all "closed", and you're kinda back to learning how to read the schematic.

when your friend pops by, try and learn why they are doing what they are doing. There's some learning curve, but there's web sites like funwithpinball.com, pinrepair.com and pinwiki.com that provide lots of background info and eventually it'll make sense enough that you can find most problems.