A myth about electric currents - What exactly happens when sending electric currents into a wire?

Everyone knows that a magnetic field is created inside and around a conductor when sending electric current into a wire conductor. Experiments we conduct with electromagnets in Japanese elementary schools are based on this understanding, and there are many products which apply the principle of electromagnets. Since various devices using this phenomenon work as they should, we are under the impression that the principle is no doubt questionable.

Electric currents create an electromagnetic field

By observing the movement of the needle of a compass near a wire while connecting a wire to a battery and a light ball, the above statement can be easily confirmed. Although the concept of fields, such as magnetic, electric or electromagnetic, is not an easy process to comprehend, let's simplify for now that fields represent some quality of space created by electric charge or magnets.

Most people know for the fact that a magnet field is created by passing electric currents through a wire. But do they really understand? There is a huge gap between the knowledge and the understanding. Let's look closely at this question, using a simple example.

First of all, sending currents in a wire means there are the movement of electrons in the wire. It's been decided by an unfortunate historical event that currents run toward the direction that is the opposite of electrons. This fact itself is not a serious problem. Since electrons' movement explanation leads us to a complicated phase in the quantum theory, we will not discuss further this issue.

The movement of electrons is relative to how we look. If we stop and look at electrons in a wire, we see the electrons moving. But if we move at the same speed as the moving electrons, they should look static. Since their movement is not so fast, it's not an impossible task.

Now, if electrons look static from us, does that mean currents do not exist? If not, does a magnetic field exist around them? Can we say a magnetic field disappears as we move?