I believe you're talking about PS2 OPL.
OPL stands for Open PS2 Loader, it's a way to put game files (isos) on HDD. That HDD would be connected to the PS2, you'd be able to play on it and have faster loading times.
You can also use an USB key instead of the HDD, though this will lead to very slow loading times (worse than legit copies).
Now, the big question : is it allowed ?
In most cases, probably not, however you should still ask the game's community about it. Due to faster loading times, it's pretty unfair, especially for games that does not provide an "Install to HDD" option.
Afaik, this option isn't common, thus it'll be considered unfair for most PS2 games.
And now, the other big question : Emu or Console ?
The answer can vary based on peoples' point of view, but in my case : If a game has an "Install to HDD" option (for example, Resident Evil: Outbreak File #2), i think it could be considered as Console run, though the comment on that run should mention it being played through OPL (ideally, maybe even add a filter to differenciate "legit" options and OPL).
If a game does not have the "Install to HDD" option, and if it is decided that OPL isn't banned from the leaderboard, the run should be considered Emulator (if the leaderboard separates Console and Emulator into different categories, like how Haunting Ground does it for example, it would be ok, because emulators have other advantages compared to "legit" options, however because they're all gathered under the same category, it matters less, because they're not being compared to "legit" options that way). Again, the comment on that run should mention if it was played through OPL.