I watched several speedrans and some players start the timer as soon as they shoot the "start" button and others when the countdown on the scoreboard ends. At what point do I need to start the timer?

Have you even read the rules? It clearly says the following: "The timer starts when the in-game countdown ends.", meaning as soon as the countdown timer in-game hits 0 and the first dummy spawns, you start the timer. Please read the rules before asking next time.