What is the name of the current produced when a conductor moves through a magnetic field?

The current produced when a conductor moves through a magnetic field is called induced current.

This is an example of electromagnetic induction: a changing magnetic flux through a circuit creates an electromotive force, or induced voltage. If the circuit is closed, that voltage drives an induced current. The motion can come from moving the conductor, moving the magnet, or changing the magnetic field in another way.

Michael Faraday demonstrated electromagnetic induction in 1831. Lenz's law describes the direction of the induced current: it opposes the change in magnetic flux that produced it. This opposition is why generators require mechanical work to produce electrical energy.

Eddy currents are also induced currents, but they circulate in loops within solid conductors rather than necessarily traveling through an obvious wire circuit. Displacement current is a term in Maxwell's theory associated with changing electric fields, not ordinary charge flow through a conductor.