Usually Stable Diffusion (and most machine learning models) run on a server with the web front end (eg A1111's SD web ui) just providing a user interface. Even if you download it, you need to run the server on your computer. When you make a request to it you are using the CPU/GPU on the server.
The linked version runs a version of the stable diffusion model in the webbrowser, so it uses your own CPU (and in the case GPU) via the APIs provided by the web browser. This specific implementation uses an API called WebGPU which isn't yet widely supported.
The linked version runs a version of the stable diffusion model in the webbrowser, so it uses your own CPU (and in the case GPU) via the APIs provided by the web browser. This specific implementation uses an API called WebGPU which isn't yet widely supported.