Apple says to update my iPhone 5 to iOS 10.3.4. But the phone says 10.3.3 is the current version and that it's up to date.

Year is relevant. Apple explains what models are effected in terms of year of introduction: "...some iPhone and iPad models introduced in 2012 and earlier will require an iOS update." Older models used a different chip.

Thanks again for helping me figure this out.

Thanks, ckuan--you helped me figure out that I'm okay for now. Apple's warning is about iPhone and iPad models introduced in 2012. The 5c came out in fall 2013.