Events: Remove time display on 'All day' events

Motivation

Currently events marked as 'All day' show their time as being from 12:00AM when viewing the event, which users find confusing. This is also the case for multi-day events that span several days. (See screenshot.)

Proposed resolution

Hide the time value on the display of 'All day' events. By definition and 'All day' event should not have any times associated with it. Otherwise it could have a proper start and end time (I wish all events could, that would make things much simpler!).

Challenges

I've done some research but I'm not familiar enough with how dates/times are handled yet to tackle this. I cannot see how the 'all day' flag is stored on the node? Or if it is at all?

I'm also not sure how this would work best with localisation. ie: how an all day event in one timezone should be displayed to a user in a different timezone. Currently the dates/times of the 'all day' event are shifted to the user's timezone. So the all day event:
13 - 15 February
12am
for Europe/London

becomes

12 - 14 February
9:00PM
for America/Argentina/San_Luis

One approach would be to still ditch the time, and keep the 'original/authored' timezone's dates, but then display the timezone? So:
13 - 15 Feburary (Europe/London)

But that interferes with how dates are handled based on the current user's timezone.

I suppose this makes the issue related to:
#2171709: Event Date display cannot be localized
#2155873: Display the timezone for events and calendars. Add a "Show Timezone" widget