a lot of us use what is called a capture card(i use an elgato hd60 for example) which allows us to stream the content to our PCs, rip the video and sound it into OBS and use a program called "livesplit" added on that where we sync up what we need from an xbox POV which requires 2 monitors(sometimes this is slightly out of sync as it can have about a 1 second delay to the PC), this is a bit on the more pricey end of the scale

livesplit is appreciated but NOT a requirement to have it on the screen the time of the game can be off screen, it looks better with it included on the final product but in reality we just need the base footage of the run and a highlight on twitch for it to be authenticated as a speedrun

the old games start the timer when we gain control of Marcus(gears 1/2/3) the newer ones use more of a loading method

hopefully this helps somewhat