It doesn't matter if you have a water/coolant mix or just water. Neither lubricates and will damage the engine. With that said, how long has it been since you did the last oil change and when was the last time you checked the oil before this happened? Reason being if the car has had a coolant leak from one of the radiator hoses and has been driven with low coolant the thermostat can have a difficult time operating correctly, especially if air is drawn into the system, and that could cause some temp spikes. The stock coolant temp and oil pressure/temp gauges are dummy gauges and don't show an accurate, true reading of what the temperature is doing and if there was some wild temp swings going on it could stress a head gasket as the head is heating and cooling and expanding and contracting (especially on a bi-metal engine with iron block and aluminum head).
If you had checked it recently and saw a milky appearance on the dipstick it would be a warning flag that the gasket may have been leaking water/coolant for a while. I'd be much more worried about bearing damage from the water if that was the case. If everything has been good you probably started running hot before the gauge started to go into the red, the thermostat wasn't able to function correctly, and as it got too hot the head warped and some remaining coolant mixed in.
The question is what point did this happen and how much longer was it driven like that. There's going to be two primary potential sources of engine damage you'll want to look for. One is if the coolant/water was in the oil for an extended period or in high enough quantity to cause bearing damage. If you hadn't already drained the oil you could've taken a sample and sent it in for analysis to see what the metal content looked like but it sounds like that's too late now.
From the scenario you've described I wouldn't be worried so much about the water in the oil as I would about the second possible source of damage and that's the oil temps. If the cooling system is non-operational and you keep driving the oil temperature will skyrocket and I've seen even high quality synthetic break down. When this happens it will allow the bearings to become damaged in short order.
It's tough to say what is going on without disassembly and if you're capable of turning a wrench there's a few things I'd do first to assess what is going on. Since you drained the oil already I'd pour a few quarts of regular oil in the car and then do a compression test. If you haven't done one of these before you can buy or rent a gauge at many auto part stores. One you have the gauge remove all the spark plugs, install the compression tester's fitting into one of the cylinders and then put the throttle all the way open and crank the car for a few seconds or until the gauge stops spiking upwards (the throttle at WOT will keep the injectors from firing and is also necessary for an accurate gauge reading by allowing the cylinder to freely pull air in). Because of the VCT you may want to put the throttle to WOT and crank the car for about 15-20 seconds before doing this to prime the oiling system and the VCT solenoid.
On cars with conventional head gaskets it can be common to have a slightly blown head gasket and still have apparently okay compression. The multi-layer steel gaskets will generally read lower. When the car didn't want to start you'd have to have an extremely low to minimal compression if there wasn't some other cause for the no-start. What you're looking for with this test is primarily to see if any or all cylinders are reading extremely low (mid to lower double digits for psi) or no compression, which would be a sign of a severely warped head. If the car was driven under heavy load while overheating it's also more likely pre-ignition could occur from a hot spot which could melt a hole in the piston but you probably would have seen metal pieces in the oil when you initially drained it (and you may be able to shine a flashlight down into the empty spark plug hole).
Next step depends on your risk taking level and if you want to gamble the damage was not as bad as you thought. If you want to play it safe and have already accepted you will probably need a new engine or a rebuild I'd pull the oil pan and get to one of the rod bearings. If you're in there and notice any bluing of the rods or crank (which will start to turn blue colored with extreme heat) then I'd plan on a rebuild. If externally everything looks normal proceed with pulling off one or two of the rod caps and remove the bearing for inspection. If the bearing surface is blue, significantly worn, has flakes missing, copper showing, etc. then you probably need a rebuild.
If the bearing looks smooth with no significant wear you may be okay. In that case you can lube the bearing and reinstall but I'd use new rod hardware and follow reassembly procedures in the factory shop manual. If the bearings look damaged you can then start planning to rebuild or find a different engine.
If you want to take a gamble and jump straight to the head (which would be next anyways if the bottom end checks out) go ahead and remove it. First thing I'd do is check the straightness of the head and also don't forget to check the block surface as well. If you end up using a MLS (multi-layer steel) replacement gasket both surfaces need to be extremely flat.
If you haven't bough one yet, pick up a factory shop manual and see how far out of spec the warpage is. If it's not too bad you may be able to get away with having the head surfaced. If the head is severely warped (like what could very well happen after continued operation with no coolant) you may not be able to deck the head as the camshaft bores would still be off. It's also possible that the valve stems could scuff and wear if it was operated while severely overheated and warped.
With the head off also take a look at the cylinder bores and top of the pistons. If there is any significant wear and scuffing where the cross-hatch lines from the initial honing are completely worn through that would be another sign of possibly needing a rebuild. You can also test the rings by rotating the engine over by hand until all the pistons are even and fill the bores with oil to an even level and let sit. If one or more drain dramatically faster than the others it may have a damaged/worn ring or cylinder bore. While it won't give a conclusive result of ring health, optimally it shouldn't leak down much if anything over an extended period.
At this point you may want to take the head to a reputable machine shop and have them inspect it. They should be able to tell you if it can be salvaged. Make sure to take the cams with you if you've removed them so the bearing journals can be inspected. If the head can't be surfaced you're probably going to want to start looking for a lower mileage used engine. If you can find one and it's a decent price it may be less expensive than finding a different or new head and valvetrain and then potential rebuild costs if the lower end was also damaged.
Best case lets say you do check the bottom end and it seems to show no signs of damage and the head isn't warped too bad and can be surfaced. You might get lucky and can install a new head gasket and be fine. If you end up going this route I'd immediately do another oil change after running for a short period in case any water or contaminants were left in the engine and then after 500 miles or so take an oil sample and have it analyzed for wear. There quite a few places that can do this but I usually recommend
Blackstone Labs. This will tell you if there was any other internal damage and ongoing wear issues. If a bearing was damaged it's going to continue to degrade. I'd probably repeat this again at a couple thousand miles and then at the next oil change just to be safe.
It sucks and I'm sorry to hear this happened but just take your time and make a logical plan on how you want to attack diagnosing and
then you'll be able to choose a repair path. Hindsight is 20-20 as they say but hopefully someone else reading this will be motivated to fix a small issue that could come back to bite them (i.e. persistent coolant leak) and for everyone else it's a good reminder to shut the car off and pull over as soon as safely possible if the temp gauge ever pegs or the oil light comes on. Even that can't always save you but it can sometime help minimize the repair bill.