According to our captain's log timestamps, its 2013.
Correct. STO's stardate counter started at 2409 when this game was in beta in 2009 and has kept ticking ever since. While technically the game itself has made no specified progression in time in regards to story elements, if you go by the stardate in your Captain's log, it's 2413 now.
Originally Posted by sophlogimo
Personally, I think it must be somethng like 2429 by now... I mean, I went from Ensign to [s]Vice Admiral[/s] Captain, and that usually takes more time than a few months...
Alternatively, our careers startet some time in the 2390s.
As I've posted in several places before when describing how to revamp the tutorial(s) and the rank progression, this game makes a whole lot more sense if we start out as Commanders in the tutorial, where we get field promoted to Captain, and stay Captain from levels 1-50 (especially since this game is about being a ship captain and there is no flag officer gameplay).