Nice video. At 0:50, I can see that the solenoid is pushing the magnet away to right, but at 1:03 (when the magnets passed the midpoint of solenoid), it is pushing the magnet to the left. Even thought the magnetic field is pointing the same direction, we have two different directions of force...
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My take is this.
The magnets are conductive.
And any given instant, the magnet completes an electric circuit composed of the battery and a section of the coil (toroid) thereby forming a solenoid around the battery-magnet assembly. The assembly is in the center of the solenoid. We know that...
