Webtiles is laggy... more so the further you are from the server too (I think).
Glyphs are also easier to distinguish very quickly, which is important if you don't want to die even more than you usually will when you try this kind of speedrun.
ophanim is the fastest human-only run (31 seconds quicker than my own :()
In my experience, strong melee characters (GrBe/MiBe are good choices) are best for this. You will be pressing o and tab most of the time, and you should practice very quickly picking up/wearing/dropping items and otherwise interacting with your inventory, and also going to Lair as quickly as possible (it is possible to reach Lair in 1-2 minutes, though not consistently, but this kind of run can't really be consistently done).
Other players also get great times with SpEn. Looking at 78291's games for instance it should be quite possible to break the record with SpEn as well, though I personally would stick with berserkers.
It is extremely important that you set your explore and travel delay to -1 in the settings to allow instant autoexplore and travel.
I'd say it's very likely that these are the fastest runs. If your internet connection is fast enough, playing locally shouldn't give you a benefit (as your own skill is more likely to be a limiting factor) and a local speedrun probably wouldn't be taken seriously anyway.