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I've moved your question from General Tech Help to XK8/XKR forum. This is the place to post tech questions about your model. Members here with the same model will be able to help.
The XK8/R is famous for mystery coolant leaks. Best solution is to 'rent' a coolant system pressure test set from you friendly, neighborhood Advance/AutoZone/O'Rilley's and pump it up cold.
I have been around this car and this board since 09, don't ever recall there being a coolant leak related to the AC being on.
Being an 01 Jag I suspect you prob have never done any work to the cooling system and prob do not know any prior maint history of car before you got it?
The cooling system on this car is vast, ha ha ha, I think.
Two radiator hoses, lower/upper, tstat housing, water pump, TWO hoses going UNDER the intake called Valley Hoses, then the octopus hose that goes around the back of the motor that connects to stuff on both side of the back of the motor, then their is the reservior seen in your pic.
I see a lot of ORANGE dried up fluid, these car come with orange rad fluid, you have or had a leak above that, for some time by the looks of it. In that area (above) there is a plastic T that goes to a valley hose, to the TB, and to the octopus hose. Typically that T goes back. You could have a leak from that reservoir.
Frankly your car being as old as it is I would get that pressure tester and make it leak, that will tell ya your leak point.
Again, I see no way that the AC causes a coolant leak, not possible.
BRGJAG surprisingly enough it is factory on both the XK8 and the XJ8. You can remove it pull the block out of its holder and adjust to take out cable slack and then place it back with a new cable tie. Right out of JTIS.
As to the leak, where do you see water when the AC is on? If in the cabin it is the drain line plugged up on the evaporator box.