My following descriptions of fields will use the term 'field lines'. Be it understood that these lines do not actually exist. They are just a way of expressing differences and variations of density, height, etc., like height/altitude lines on a topographic map.
An electric field exists with or without electric current. A static electric field exists between any two points of electric charge, be it the static electric charge between your hand and the doorknob you are about to grab or the two terminals of a battery. 'Static' generally means 'not changing or no current flowing'.
The only difference between static electricity and 'electricity' is whether current is flowing or not.
The above electric fields exist before and during current flow or current would never flow.
Magnetic fields only occur when there is current flow - or - electron(charge) motion. Such is believed the cause of a static magnetic field related to a permanent magnet. Most of the electron motion is aligned and going the same way so each of their tiny magnetic fields sum to what you perceive emanating from the magnet. You can't measure these small currents because it isn't currently possible to measure the current that which is an electron in orbit around the atom's nucleus. At least, not with common equipment.
So, all electric & magnetic fields are actually each part of an electromagnetic field where there is current flow. I will go as far to say that a static electric field would not fit under 'electromagnetic'.
Electric and magnetic are opposite sides of the same coin. Magnetic can't exist without the electric. (Common explanation>Electric can and does exist without magnetic because magnetic is relative to motion.) My belief is that neither can exist without the other. The only reason we do see electric fields without magnetic fields is because there is no relative motion between us and the moving charge. We are moving together.
The difficult fact is that there is no reality in anything being at rest from all reference points.
I'll break that thought as it is the edge of the proverbial rabbit hole.
Think of it using this analogy:
Static electric is a boat having the potential to move but setting still on a calm lake. Magnetic is the wake left by that same boat when it is moving. The question not answered easily is what is the electromagnetic equivalent of the lake water in that analogy?
I think the current trend is to say the lake water is dark matter/energy and everything is floating in that. Of course, dark matter/energy wind up just being the fudge factor to make the current math come out right. They would never admit that aether is just as good an answer.
Wiki has good explanations of both with easily understood graphics. I'm a picture guy, myself. As long as it has pictures I can understand it
"As far as the laws of mathematics refer to reality, they are not certain; as far as they are certain, they do not refer to reality." - Einstein
"What we observe is not nature itself, but nature exposed to our method of questioning." - Werner Heisenberg