I'm wondering about the reliablility of the Display.sync() method. Javadoc says "Best sync method that works reliably". Does that mean it works reliable on all systems (Windows, MacOS, Linux)? Or should I implement some fallback-code for timer-based movement/animation?

it should be fine for all of the platforms.
There are 3 versions, cunningly named sync1, sync2 and sync3

They all do the same thing - they just have slight differences in their behaviour when certain conditions are met. Please check the Javadoc.