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Electricity and magnetism are the exact same thing. A non moving charge creates an observed static electric field. A moving charge creates an observed magnetic field. So if you were moving next to the charge as it went down a wire, you would see no charge moving. No charge moving means no magnetic field.
They only look different because things get distorted moving at near light speed relative to you. It would be like if we gave two different names to a police siren based on whether you were listening standing next to it or listening as it zoomed past you.