Electricity and magnetism came together in the successfull mathematical model of Maxwell's equations. Before that, they were modelled as separate phenomena . These equations, classical electrodynamics, were very successful and have led to the technological civilization we are enjoying at present.
With the studies of the microscopic framework of atoms and molecules and particles, the theory had to be quantized , and it is called quantum electrodynamics. The two frameworks for describing electromagnetic effects, are consistent, the classical description emerging smoothly from the quantum mechanical.
Your question is answered by Arnold Neumaier within the quantum framework.
Here I want to address electromagnetic radiation within the classical electrodynamics framework.
Maxwell's equations are differential equations and their solutions for the electric and magnetic fields also give a wave solution.
Electromagnetic waves can be imagined as a self-propagating transverse oscillating wave of electric and magnetic fields. This 3D animation shows a plane linearly polarized wave propagating from left to right.
Note that the electric and magnetic fields in such a wave are in-phase with each other, reaching minima and maxima together so , even though you are correct that changing electric fields generate changing magnetic fields, and vice verso, the organization of the wave is not causal. The cause is the source and after the wave leaves the source the correlations/phases are fixed depending on the type of wave. It is interesting to read the history of electricity and magnetism .
Working on the problem further, Maxwell showed that the equations predict the existence of waves of oscillating electric and magnetic fields that travel through empty space at a speed that could be predicted from simple electrical experiments; using the data available at the time, Maxwell obtained a velocity of 310,740,000 m/s. In his 1864 paper A Dynamical Theory of the Electromagnetic Field, Maxwell wrote, The agreement of the results seems to show that light and magnetism are affections of the same substance, and that light is an electromagnetic disturbance propagated through the field according to electromagnetic laws.
At the time, even though the solutions of the Maxwell equations did not require a medium for the light, physicists used with waves from acoustic to water ones, proposed that the waves of light moved on a medium called luminiferous aether. Experimental data , the Michelson Morley experiment, showed that the aether did not exist. So it is an experimental fact that light does not need a medium to propagate.
One thing that I'm surprised to know is that light is also called an electromagnetic wave.
Does this include light of any kind, for example: light from a bulb, a tube, and also from the Sun?
Yes light of any source as visible and also radiation that is invisible to the human eye, as xrays, gammas, infrared etc .
How do they contain electric and magnetic fields?
The mathematical description contains an electric and magnetic field because Maxwell's equations are expressed in terms of electric and magnetic fields , and thus the solutions that have been identified with light and all the rest of radiations, also have the electric and magnetic fields in the functions.