I think your machine will have a cooler like in the picture. Engine side,below the radiator.

Water hoses go on each end and oil hoses to the small fittings.Water circulates through the outside of a matrix of tubes which are filled with transmission oil.If the matrix leaks,water gets in the oil but tbh,I'd maybe think oil could then get into the water system as well.
Do you have a cooler like above or is there a second radiator in front of the main coolant rad? If there is one in front,it'll be full of oil and is cooled by the fan and has no connection with the coolant.