FWIW I have the ASUS ROG Swift PG32UCDM 31.5" 4K UHD (3840 x 2160) 240Hz Gaming Monitor - a $1.3k monitor and also don't see anything different in the demo. Maybe I'm looking at the wrong thing. https://www.shadertoy.com/view/XfKfWd
Make sure your browser lets you refresh at those framerates.
Safari by default caps animations other than scrolling at 60fps (I think?).
This will let you know what your browser is allowing: https://www.testufo.com/framerates (it will also demonstrate the issue that the post is attempting to solve).
I only get an hyphen on the iPhone
I got 60fps/60Hz on mine (iPhone 16 Pro, iOS 18.2).
Just FYI, you can go to Settings -> Safari -> Advanced -> Feature Flags -> Prefer Page Rendering Updates near 60fps and switch it off to get 120Hz
240Hz is more recommended, 120Hz does not give much benefit, especially on small screens. If your screen is bigger you may notice, but it starts to become noticeable if done on OLED instead of LCD, or if you increase refresh rate by 4-8x above CRT simulation target.