I've noticed some strange behaviour with this too and my conclusion is the following:
If you set the start date in any other month than June, once you leave the screen and come back it will have increased by one.
Further, it appears there is always (even in June) a discrepancy of 1 compared to what the editor shows, and what the mission shows when you fly it.
Example:
The editor thinks it's day 62:
launch the mission, now it's day 63:
Same thing if you leave it on day 0 (june 1st).
editor 0:
in game day 1:
Still, for any conditional behaviour to work, it seems you set whats in the editor, not what the game says (example, want a group to orbit until time 1000 on the same day. The editor thinks day 11, set that. Even though the game will show day 12. If you set orbit until day 12 1000 theyll hold the whole day)
It's a bit of a strange discrepancy, would be nice to be fixed.
Tested on two computers both running the latest version of 1.2.5.16508