The autosplitter that appears by default in LiveSplit is only for the 6th gen PC version of this game. I'm not familiar with Playstation emulators, but 2.0.5 should be fine as long as there aren't any major issues with the way it emulates the game.
According to this list of recommended emulators for THPS games (https://pastebin.com/3MdnwLph), you'll be wanting to use ePSXe version 1.9.25 or 2.0.0 for your runs. The 5th gen leaderboards do not currently use in-game time, so you will only need to submit real time.
📞 Yo! The guy just called and said he's in trouble over by the Slam Bros. Warehouse! Check it out! 📞 What?! Somebody jacked my camera? Man, you better find that guy! 📞 Are you kidding me? He tossed the film? Head for the camera shop in the plaza. 📞 Perfect! Now get back here with the kit before these guys bail!
For IL submissions, you only have to enter your time into "in-game time" because the "time" column just makes things redundant for the IL boards since it's just the same thing. For full game submissions, "time" is determined from the moment you start the first level to when you get your medal in the final level, and "in-game time" is the same thing except the time spent in loading screens is subtracted from that time in order to level the playing field for runners who play on slower hardware. If you plan on doing full game runs, there are other rules you should also see by going on the leaderboard and clicking "view rules".