Which is the best way to produce electricity?
First of all, by "best way" I assume you mean the most efficient way for the same input power of the DC motor. 

The physical law for generation of electricity, here, is Faraday's Law. Faraday's Law states that electromotive force generated on a conductor is proportional to the rate of change of magnetic field on the conductor. According to this law, there is no difference between moving the coil or moving the magnet; they have the same effect.

Your question is more about the practices, i.e. setting up the actual circuitry. Your both proposals of current generation will result in an alternating current. In the first method you proposed you can use commutator to switch this alternating current to direct current. But in the second method you proposed, you cannot use the commutator, therefore you will have alternating current. Actually, you may come up with a commutator design but it should be more complicated than that are used for rotating coil arrangements.

In the end of the day, both methods will result in same efficiency-assuming every other effects are the same. Your decision should be on whether you want alternating current or direct current.