Cool. I respect PhD in Physics (and scientific education or background in general).
Yes, I would say that what we call electric field is NOT only just imaginary lines of electric force with density decreasing from a point charge as inverse square of the distance in 3-D space (inverse first power of distance in 2-D and uniform non depending on distance in 1-D).
This object (electric field) also has energy with the density epsilonxE^2/2 hence inertial and gravitational properties as any other kind of energy does.
When charge in respect to non moving observer observer, space in the direction of motion for such observer contracts (relativity theory), thus the electric field density and the direction of E vector becomes different than the E of static charge. (They call this situation no longer electrostatic, but electrodynamic). Due to higher density of E you have more energy in this field now as well. And this difference between static field and dynamic field is what we historically call magnetic field. It is small for slow motion, but becomes essential when you move with the speed close to speed of ligh. We would not notice the small difference at all if not the fact that electrostatic force is extremely strong one but the matter around us consists of equal number of opposite charges thus the total average field of static charges of non-moving protons and moving in random direction electrons is practically zero. But if electrons start moving orderely (current), then due to small relativistic difference in the field of moving charge versus field of non-moving even this small difference immediately becomes noticeable, and we call this difference "a magnetic field". So, the magnetic field is a non-compensated by opposite charge relativistic part of electric field of moving charge (non-compensated because protons are NOT moving).
To make an example, calculate force of repulsion between two moving charges using relativistic formulas, and you'll find it to have a relativistic factor sqrt(1-(v/c)^2)in denominator.