General Caption: Note the old IN/IS dichotomy or the single meaning for the two locutions:

1. Electromagnetic radiation (EMR) is a traveling, self-propagating, transverse wave state IN the electromagnetic field which is everywhere always in all spacetime---as illustrated in the images. This what is one says when being very exact.

2. Electromagnetic radiation (EMR) IS a traveling, self-propagating electromagnetic field---where one means by electromagnetic field the particular electromagnetic-field STATE constituted by the particular set of traveling, self-propagating EMR waves that you are considering.

1. The first locution is exactly true, but the second is what people usually say---it just easier to say: i.e., electromagnetic radiation (EMR) is a traveling, self-propagating electromagnetic field.
2. The EMR is independent of source or sink.
3. To generalize, one almost always says electromagnetic field meaning a particular electromagnetic field state. mutatis mutandis for electric field and magnetic field.

Illustrations of the electromagnetic field which is everywhere always.

1. Image 1 Caption: Diagram of electric charge (positive charge (+) and negative charge (-)) arrangement in oddly shaped electrical conductors and the overall electrostatic field both caused by the fixed placement of an ideal point charge (a with positive charge (+)).

   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.

   

2. Image 2 Caption: "An alternating current (AC) through the solenoid, producing a changing magnetic field. This magnetic field causes an electric current to flow in a wire loop by electromagnetic induction" (see Wikimedia Commons: File:Electromagnetic induction - solenoid to loop - animation.gif: slightly edited).

   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.

   

3. Image 3 Caption: "An animation of a half-wave dipole antenna (a stardard radio antenna) emitting radio waves (radio band fiducial range 3 Hz -- 300 GHz = 0.3 THz, 0.1 cm -- 10**5 km) showing the electric field lines ... " (see Wikimedia Commons: File:Dipole xmting antenna animation 4 408x318x150ms.gif: slightly edited).

Images:

1. Credit/Permission: Chris Burks (AKA User:Chetvorno, 2010 / Public domain.
   Image link: Wikimedia Commons: File:Electrostatic induction.svg.
2. Credit/Permission: © User:Ponor, 2005 / CC BY-SA 4.0.
   Image link: Wikimedia Commons: File:Electromagnetic induction - solenoid to loop - animation.gif.
   
3. Credit/Permission: User:Chetvorno 2015 / Public domain.
   Image link: Wikimedia Commons: File:Dipole xmting antenna animation 4 408x318x150ms.gif.