Is it safe to keep running a split system that's dripping water inside?

Nine times out of ten this is a blocked drain line, and no, the unit is almost certainly not ruined. The water dripping out the front vent is the giveaway. Your indoor unit pulls moisture out of the air when it cools, that water collects in a little tray underneath, and it drains away thru a small pipe that runs outside. When that pipe gets clogged with gunk or slime, the tray overflows and the water spills out the front, exactly where you found it. Cooling fine, no noises, no smells means the guts of the unit are in good shape. Water damage from an overflow is ussually cosmetic on the wall and nothing inside the unit itself. Six years old is about the age where this starts happening. Algae and sludge build up in the drain line over time and eventually it plugs. It's routine maintanence stuff, not a death sentence for the unit. wheather you DIY it or call a tech depends on how handy you are and where the drain exits. If you can find where the pipe comes out outside, some people clear it with a wet vac on the end of the pipe or flush it through gently with water. A tech can do the same job in half an hour and also clean the tray and check the coil while they're there, wich is worth doing on a six year old unit anyway. Either path works. On the fire and electrocution worry, I get why it spooked you, but the electrics in these units sit up top of the indoor unit and the tray sits below them, so an overflow genrally drains away from the electrical parts. That said, you did the right thing switching it off, and letting everything dry out for a day or so before restarting is a sensible move, expecially with water having pooled on the skirting below.

One more thing to check while it's off. If the dark patch on the paint has spread or the plaster feels soft, keep an eye on it, since sitting water on a wall can cause mould behind the paint down the track. Airing the spot out now will save you a repaint later.

Same thing happened to mine a couple summers back, and it turned out to be exactly what was described above. I put a wet vac on the pipe where it exits outside and it sucked out a chunk of black sludge, then the water started flowing freely again. Took about ten minutes and cost nothing. Mine has clogged twice now in eight years, so once it happens it tends to happen again. I flush mine with a jug of warm water every spring now and have not had a repeat since. Cheap insurance for a ten minute job.

That makes me feel better about mine actually, it did the same thing last winter and I panicked and rang someone straight away instead of even looking outside first. Paid for a call out when I probly coud have sorted it myself with a bucket and a hose. One thing I still don't get though, when you go looking outside, how do you know which little pipe is the drain one? I've got a couple of white tubes running down the wall and no idea which is which.

Not sold on the idea that the electrics are always safe becasue they sit above the tray. On my old unit the wiring and the drain pipe both run down through the same little hole in the wall behind the indoor unit, so when the tray overflows the water can track back along that penetration into the wall cavity instead of just spilling out the front. Mine did exactly that and I ended up with a damp patch well away from where I expected it. So yeah it is probably just a blocked drain, but I whould not assume the water went nowhere near the internals just because of where the tray sits.

Fitting these for a living, and the thing I see over and over is the drain pipe sagging or back-pitching somewhere along the run. Six years is classic for it, but on a proper install the pipe shoud drop consistently the whole way out, and I always set mine to about a 1 in 50 fall. If yours keeps blocking every year or two like the other poster said, it might not just be sludge, it might be water sitting in a low spot growing that gunk. Worth checking with a spirit level along the exposed pipe outside. If it sags, propping it up and re-sealing the entry hole behind the unit sorts most repeat offenders.

Update: turned out to be a blocked condensate line, just like the first reply said. I found the outlet outside, ran some water backwards through it with a hose and a load of gunk came out. Turned it back on and it's been running all evening with no drips. Now I just have a stained patch on the wall to sort out. Kept the power off untill it was dry anyway, which seemed sensible given what that last reply said about the wiring hole.