Sounds like what you have may be pink eye, which might very well be spread in another manner - on our first family trip to DL in 2006, our daughter had a nap 2 afternoons prior to us leaving on our flight to DL, and her one eye was totally RED upon awakening. (No prior signs or symptoms at all.) It was pink eye, the Dr. told us, and was easily treated with drops. I hope yours goes away quickly, and that no one else gets it. I'm not dismissing the idea it could have come from the water, but I think it is more likely to be pink eye than an infection from the water. Because the water does hit people's faces, Disney likely has to chlorinate pretty heavily, and test the water (I would assume.)
Edited to add: Even if you're good about not touching your face, etc... it takes one second for you to rub your eyelashes if it feels like something's in them, etc., to place bacteria or a virus in that area, and a person really doesn't even know they've done it. They've done chalk tests before where a person's hands are covered with pink chalk (on the palms) and before long, people have touched a lot of areas on their faces.
I'm curious to know how clean the water is on Splash and on PotC. It didn't smell offensive to me on either ride, it just smells nice and humid, like when you walk onto the POTC ride que, just as you get indoors. LOVE that smell! Miss it so much!!