Dear
In my opinion, all fields exist, they are not generated or induced by the charges (in general). The charge is a measurable physical quantity giving the coupling of the particle with these fields. It did not induce the field. I invite you to think about how particles acquire an electric...
In fact the electric field and magnetic field are parts of the same object "electromagnetic tensor" so in a frame where the charge is at rest we see only the electric part but in an other frame where the charge moves we see the two parts. I hope that helps
Thank you