General Caption: Note the old IN/IS dichotomy or the single meaning for the two locutions:
Illustrations of the electromagnetic field which is everywhere always.
The electric field lines are a visualization tool that at every point align with the electrostatic field at that point: the arrows give the direction of the electrostatic field.
At the macrosopic level, there is NO electromagnetic field inside the electrical conductors, but at the microscopic scale there is---it just is small scale and averages away at the macroscopic scale.
Remember the electromagnetic field is everywhere always in all spacetime.
The magnetic field lines are a visualization tool that at every point align with the magnetic field at that point: the arrows give the direction of the magnetic field.