OK, I see what you are asking now -- I haven't been following this thread from the beginning.
In the bottom right diagram, a static (relative to the frame)
electron near to but outside the lower wire will be attracted to it due to the net positive charge on the wire. As the electron is static, magnetism is irrelevant to it.
Translating that to the bottom left picture, the electron is now moving but the wire is not charged, so there is no electrostatic force. Nevertheless, there is still an attractive force, as we proved using the bottom right picture. The explanation for this force is magnetism. If you already knew about electrostatics and relativity but knew nothing about electromagnetism, this argument would effectively define for you what electromagnetism was.