For instance this timer,what is it called? It seems to be the best tool to get the exact timestamp
If you want you can always run the run through expression encoder 4 and add the timer into it and get the exact frame to start and stop too
It should be noted that unless you run Livesplit with an auto splitter it is not the exact timestamp. Without an auto splitter it's manually controlled by the streamer so it's subject to human error on starting and stopping (but it's usually very minimal).
Even then, autosplitters aren't magic. They check the game for whatever triggers it 60 times a second, which can also be de-synced with the monitor refresh rate/OBS capture rate. The margin of error at most would be ~33ms.
Edit: Actually, because LiveSplit refreshes at 40Hz, the margin of error could be as bad as ~100ms.
That is true, I should have phrased it that an auto splitter makes it more accurate than manually doing it
I used to record my console runs as .avi (using debut capture software) and open them in virtualdub, and check the time down to the frame.