Hello.

A coil with a laminated iron core, put to spin... Would it generate some electric current in its windings just because the faint earth magnetism is in the way ?   

-No permanent magnets involved-

Yes. We had a physics lab when I was a TA that detected the earth’s field this way; the students measured the voltage in different orientations.

(An interesting anecdote is that students in one corner of the room got a different answer because there was an NMR lab in the building, and the field in that corner was measurably different.)

edit: there are a few schools that list a similar lab online, such as

https://www2.oberlin.edu/physics/catalog/demonstrations/em/earthinductor.html

The device rejoices in the name of an earth inductor.
https://geomag.nrcan.gc.ca/lab/vm/inductor-en.php

Interesting contraption, thanks. 

If uses air cored coils, generation should increase with an iron core, right ?   

This thread triggered from inspecting my defunct portable generator that shows no residual magnetism anywhere at all and has no brushes.  Wondering if earth magnetism is used as initial development of field excitation... 

FC= field coil rotor   MC =stator main coil   SC=stator coil for capacitor   PL=pilot lamp