If a light bulb produces light and if this light bulb is eternally "on" (and therefore has always been producing light), it does not make the light self-existent. Its existence is rooted in the bulb.
Also, "father and son" relationship are quite clear terms regarding who is of who.
"No one has ever yet seen God. The only begotten God, the One being in the bosom of the Father, He has made Him known."
J 1:18
Trofimus,
We may just be misunderstanding each other, or using wrong terms at this point.
When I say Christ is "self existent", I am NOT claiming he is "independently existent".
I am simply making the distinction he is not "contingently existent"... meaning that something prior to him created him... which would mean he is not eternal.
For Christ to be eternal, he cannot be contingent... there cannot be something prior to hm which created him.
It seems the trinity is both eternal and coexistent.
This is something we don't really have analogies for, so it's natural to debate the finer points of this as we parse through it.
Coexistent neither means "independently existent" nor "contingently existent"... its something for which we don't really have adequate analogies.
Regarding your analogy of the lightbulb being the father, and the light from it being Christ, I think it's better if we put this analogy back into it's original form, which was using the analogy of the sun.
In that analogy, the sun is the father and the rays of light are the son.
But here we have an interesting dilemma... the rays cannot exist without the sun, but the sun cannot even BE the sun without the property of emitting rays of light.
So the SUN and the RAYS OF LIGHT seem to be "coexistent" in some way... and neither can actually exist without the necessity of the other coexisting.
In reality, neither is really contingent, but they coexist.
I don't think this analogy is perfect,
but it was originally developed in order to show "eternal coexistence"... and for this purpose I guess it's fine.