Yes you are completely right, but it took an intellectual revolution for physicists to realise that it made sense to have a wave that wasn't the motion or jiggling of some physical medium, a wave that could exist equally well in empty space.
All light is electromagnetic radiation. But to ask « How do they contain ... » is a little vague or philosophical. The correct answer is, the light is the field.
By the way, there isn't really a difference between the electric field and the magnetic field, they jointly compose one electro-magnetic field, and this one field cannot really be divided into two separate parts, one electric, the other magnetic, because two different observers who are moving in different directions would divide the same field differently, what one called electric, the other observer would call magnetic, showing that the division into two parts is somewhat artificial.
Finally, to answer perhaps what you really meant, how does the light bulb or whatever contain the field, the answer is it doesn't exactly at first, but turning on the current of electricity that flows through the filament produces motions in the electrons in the atoms in the filament at just the right speeds so that that motion of the electricity in the atoms generates a magnetic field and that starts the whole process. In its fundamental principle, it is the same as with a radio antenna, but the frequency of the motion is different so the electro-magnetic field produced is at a radio frequency instead of a light frequency, and there are other differences of detail also. I will not explain about how atoms emit photons since that, in a way, is already contained in what I said, I just said it in the wave picture of classical electromagnetic theory instead of in quantum terms.