Him timing it "down to the millisecond" may be a touch disingenuous; the time 0.567 is about 17/30, or 17 frames assuming the video is 30 fps, the default rate to export frames from a Source engine demo (see here). The final time he calculated (in seconds) would then be the number of frames he got from the
startmovie command divided by 30.
As far as how speed runs are "officially" timed, the most "official" (notice the quotes) body I know for speed-runs would be the Speed Demos Archive, who maintain a decent amount of runs across various platforms. They have an FAQ about how to time runs, which may also play a role in how he figured his time:
If a game displays a time upon completion, and this time is tested to be accurate, then the timer will be used. An example of an inaccurate game timer is one that doesn't display the exact time when a player saves, such as Star Ocean 2's timer, which drops seconds when saving. Some game timers don't count time at pause/inventory screens, dialogs, cutscenes, etc., so the time can be significantly less than the video length. Some games have a timer but it can't be seen at the end of the game. In most of those cases the timer will be ignored. For some games like RPGs, however, such timers are displayed in a menu screen that you would be looking at before the final battle; that time will be noted and real time from that point added on.
For games without timers, a simple real-time measure is used. When the player first gains control of the game's character, timing begins. At the end when control is lost, even if that's long after the final battle, the timing stops. Possible movement that can occur during or after the ending credits does not count. For segmented runs, timing for a segment stops at the first system-dependent activity, usually the actual saving. When loading, the timing resumes at the point when the game was saving or displaying the password. For runs over three hours, the seconds are dropped because slight variations in recording speed can become significant.
Note that in the linked video he times it to when he completes the final boss objectives, which is not how SDA would, so their time would be 15-30 seconds longer, with their end being when the cutscene starts. Their theory would be that there could be an (undiscovered) glitch that would speed up arriving at the cutscene, so his timing would cheat someone who uses it.