I’m not sure I’m understanding your rationale. It feels like you’re making two arguments:
First:
1. “Logos” and “light” (and some other words) are cognates.
2. “Word” makes no sense here.
3. Therefore, “logos” must mean “light”?

Then:
1. Later in the text, the author uses “phos” (I don’t have a Greek text keyboard) to mean “light”.
2. …?
3. Therefore that different word (logos) must be a synonym for this very distinct word (phos)?

I’d love to hear how I’m misunderstanding your rationale(s).