"Midnight" doesn't solve the second part of GP's problem:

> In addition to the confusion about whether 12 PM is 00:00 or 12:00, there’s confusion about which calendar day 12 AM falls on. When does a Friday midnight movie start? 99.9% of people will show up at the end of Friday, but in a literal user interface you’d have to choose 12 AM Saturday.

I use 11:59pm or "end-of-day" for most cases.