Solenoid "vibrates" instead of pushing
From your schematics it looks like:


you connected the power supply backwards (on both schematics actually, I wonder how come you did not fry your Arduino)
the external supply and the Arduino do not share a common ground as they should.


Edit: about the orientation of the power supply:


the (−) side of the supply (and GND of the Arduino) should be connected to the source of the N-MOSFET
the (+) side of the supply should be connected to the cathode of diode (the side identified by the small white bar)