What are the differences between these two words? Are there any contexts in which one would be appropriate while the other would not? It would be particularly illuminating (pun intended) if there were a passage in which the two were used side-by-side in contrast to each other.
I did a search for lux and lumen in the Latin dictionaries at Latinitium https://www.latinitium.com/latin-dictionaries?t=lsn27193,do307 They seem to overlap in meaning, with lux being used more often for daylight and lumen being used more often for lamps and torches. The entry for Doderlein at the bottom of the page has examples where they are used in contrasting ways, and says
Lumen (λευσσόμενον) is a luminous body, like φέγγος; lux (λευκή) a streaming mass of light, like φάος. Cic. Fin. iii. 14, 45. Ut obscuratur et offunditur luce solis lumen lucernæ. Curt. viii. 2, 21. Sed aditus specus accipit lucem; interiora nisi allato lumine obscura sunt. Cic. Acad. iv. 8, 28. Si ista vera sunt, ratio omnis tollitur quasi quædam lux lumenque vitæ; that is, reason alone is in itself bright and light, and at the same time spreads brightness and light over life. Also, in a figurative sense, lumen denotes distinction, lux only clearness. Cicero (Man. 5.) calls Corinth, Græciæ totius lumen, but Rome (Catil. iv. 6.) Lucem orbis terrarum; Corinth is compared to a glimmering point of light; Rome is distinguished as that city in comparison with which all other cities lie in darkness. (ii. 66.)