A particle with spin, such as an electron, has two different magnetic fields, one due to its motion through space and one due to its spin. (Note: The spin of an elementary particle can not be understood as that particle simply rotating in space, like the Earth spins on its axis. It is not a kind of “motion through space”.)
The magnetic field due to its motion through space can be considered a “relativistic effect” when the electron is moving uniformly. In the electron’s rest frame, this field completely disappears, and in other frames the magnetic field can be found by making a Lorentz transformation of the purely electrostatic field in the rest frame. So the magnetic field of a uniformly moving charge can be seen as a relativistic artifact of being in the “wrong frame”.
However, an accelerating electron has a magnetic field which cannot be made to disappear by choosing a particular reference frame. It is not the Lorentz transformation of some purely electric field. The magnetic field of an accelerating charge is not a relativistic artifact of being in the “wrong frame”, so you should not consider it a “just a relativistic effect”, even though electromagnetic radiation cannot be properly understood without relativity.
The magnetic field due to its spin is an intrinsic quantum-mechanical effect that cannot be made to disappear by choice of a reference frame. The magnetic field due to spin is not a relativistic artifact of being in the “wrong frame”, so, again, you should not consider it as “just a relativistic effect”, even though the proper understanding of spin does involve relativitistic quantum mechanics.