It's correct, but I think it's misleading. If you're now thinking that magnetic fields are just electric fields viewed from different reference frame and that t

It's correct, but I think it's misleading. If you're now thinking that magnetic fields are just electric fields viewed from different reference frame and that the latter are the more fundamental, that's not right. There are situations in which you have both fields and you can't attribute the magnetic field to a purely electrostatic field in another frame. It turns out that all observers agree on the value of k=E²/c² - B², so if you have k<0, you can't possibly find a frame with no magnetic fields, because that would imply k≥0. So, you can use relativity to motivate the need for introducing a B field, instead of "pulling it out of the hat", but in general you need both E and B. In the modern formalist these are the components of a larger object called F, the field strength tensor.