I don't know what cursor styles mean. As I said, adding pointer trails fixes it but there's two things which makes this not a solution at all: Makes the cursor feel very lag-like and you cannot do it in Dota if the game is running in fullscreen.
Control panel > Mouse > Pointers tab > Scheme dropdown menu. I haven't played LoL for a while but check if it has an option to disable Mouse Hardware Acceleration in Options.