Here's a fun ticketing wonk phrase to add to your arsenal: "Late-night disambiguation". Late-night disambiguation is the art of making it clear to end-users exactly what day you're talking about.

Whether you have your own custom website integration (through the External Web API) or use the Elevent for WordPress plugin, instead of displaying 12:00 AM on the following day (technically correct but who has time to think about what that means), we'll display any midnight times as 11:59 PM--the last minute of that day.

While less technically correct, it's worth it to be a minute off to communicate clearly to end-users.