In electromagnetic induction, does the direction of the magnet’s motion influence the direction of the induced current?

Yes, it does.

In fact, Faraday-Neumann-Lenz law can be written as

where
is the induced emf
is the magnetic flux variation
is the time interval

The negative sign in front of the formula gives the direction of the induced emf. In fact, the direction of the induced emf is such that it opposes to the change of magnetic flux. In other words, if the magnetic flux is increasing (for instance, when a magnet is moving toward a coil), the direction of the induced emf is such that it produces a magnetic field opposite to the original magnetic field; viceversa, when the magnetic flux is decreasing (for instance, when the magnet is moved away from the coil), the direction of the induced emf is such that it produces a magnetic field in the same direction of the original magnetic field.