With the Fermi energy at 6.1 eV, the Fermi surface is more complicated, but the description of the conduction process is similar. An electric field directed to the right shifts the electron distribution to the left, and there is a net electric current to the right. But if you stop to count the number of electrons that are outside the Fermi surface it is rather small because there's practically "no Fermi surface available for the E-field to push the electrons across." The argument becomes more apparent if you add electrons by shifting the Fermi energy to 7 or 8 eV. Obviously, we can no longer expect the current to be proportional to the number of electrons.
If you turn on a 5 Tesla magnetic field, the patterns are intriguing but NOT easy to interpret. (The green dot gives the average k of the electrons, but is of little physical significance at energies for which the Fermi surface touches the Brillouin zone boundary.)
For an alternative interpretation of this situation, open preset 4.