Honestly this confused me too when I was buying my first projector. Lumens measure total light output from the projector itself, while lux measures how bright that light appears on the screen - lux depends on both lumens and screen size. So same projector will have higher lux on smaller screen and lower lux on bigger screen. For comparing projectors, always use lumens since that's the consistent measurement regardless of screen size.

I use an Optoma projector and when I was evaluating, I focused on ANSI lumens specifically - that's the standardized measurement. Some cheap brands advertise "LED lumens" or "peak lumens" which are inflated numbers, not real brightness. For dark rooms 2000 ANSI lumens is fine, ambient light rooms need 3000+. Don't get fooled by high lux numbers on small demo screens in stores - always ask for ANSI lumens rating before deciding.