This one is basically my daily driver for similar tasks: https://app.heynote.com/ (full app at https://heynote.com/). And it seems to work almost out-of-box for your first two examples, once you switch the buffer from "Plain-text" into "Math" mode.
It also supports switching between different buffers and some kind of local storage.
It also supports switching between different buffers and some kind of local storage.