How do you proof of the existence of back emf ?

yes..only the application is different.

yes indused emf is also called motional emf.

If an open coil is subjected to a variable magnetic field, at the ends of the coil a potential difference is induced which is called induced emf.

If a coil is connected to an emf source and switched on, the rising current will produced an variable magnetic field which in turn produces an emf. It is called back emf.

It is also called back EMF. Back EMF is the electric potential difference that opposes the current that induces it. See related link below.

In dc motor, the armature conductors are revolving in the magnetic field and emf is induced in the armature conductors. The direction of the induced emf is in opposite direction to the applied voltage as per Flemings left hand rule. So, the induced emf in motor is called as back emf or counter emf.

Vydehi