I don't think there's any problem using water putty on concrete providing it stays dry.
Water putty is what they use to fill knots in the plies of plywood. It's water resistant, but not water proof. Also, it's not super strong.
I think what may have happened is that the water putty may have shrank a bit as it dried, and that allowed some wiggle room for the toilet to move. If the water putty was in continuous contact with water, I expect it would have softened up, allowing the toilet to move more and more and more until it started to leak. I'd have a plumber look at the toilet floor flange and see if he can find a reason for it to have started leaking.
The purple/brown discolouration that you see inside linoleum is actually mold or mildew growing in the paper backing of the linoleum. That means the paper backing of the linoleum has gotten wet. To my knowledge, there is no way of cleaning or removing this discoloration besides replacing the linoleum.