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One thing to consider about the moving electron is that the E field lines do not bunch at the point perpindicular to its motion and continue off into space in straight lines. If this were true, then the field from an electron would "move out" instantaneously. Rather the field lines resemble fluid streamlines, they curve opposite the velocity vector. The curvature is proportional to the speed.
Now, if they curve opposite the velocity, what would be the resultant E vector? Its divergence.
If the moving electron is considered a current of density J and it is moving at a high velocity so almost all the E field lines are perpindicular to the motion, then the work done on this particle: Integral E dot J is zero. The work done decreases as the velocity gets increases. Reliable model isn't it? Perhaps the field lines are confused with the dimensional Lorentz contraction of the electron.

If an electric field causes charge to move (the charge moves to establish an electric field of its own that counters the impressed field), then why is this field of the charge absent when the particle moves due to forces other then electric fields? The field exists only in the vicinity of the charge.

I'm trying to find a reference for you on this that is not proprietary. There may be something in electrohydrodynamics, frictional electrification or in the atmospheric sciences.

Physicist:
If the E field lines of a moving charge do bend opposite the motion to form "streamlines" then I have to concede that an additional E field does not appear when the electron moves. What happens is the existing coulombic E lines rearrange themselves to oppose an applied field (by moving the charge) and for motion not due to an external field the lines rearrange to give an net E vector opposite the motion. The charge still meets Gauss's law and other observations. To me this is the simplest and most intuitive explanation and the more I think about it the better it sounds.
You may still believe the field lines are rigidly attached to the charge and extend as straight lines outward as the charge moves, but I don't think it's so.
Anyway, you are right that no new E appears. To me its simply a natural shift of the radial field to some other geometry in response to motion. Anyway, thanks for the head banging, I owe you a beer.