I've noticed that in some speedruns, the player is able to trigger the end of the level after killing only 4-5 Tyrasaurs instead of the standard 10. Can someone explain to me what needs to be done to trigger the earlier ending?

I think it just depends on whether you're playing on the Japanese version or the US version. In the Japanese version you just need to despawn a certain amount of them to trigger the ending (although some are still killed for the exp), while in the US version you need to kill 10.