however deleting the whole obj/ folder and the executable is more than enough (I think it is exactly what the "clean" do).

deleting part of the source is not needed because the message

mingw32-make: Nothing to be done for `all'.

appears when the compiler founds all the files in the obj/ directory and no newer source file. When you delete obj/, the compiler will compile the whole source again because it finds nothing in the obj/ folder