The browser version/demo is still free - and has all of the content you remember from the free version. The full version just has a lot MORE content after the end of the "demo."
No, the demo already pushes the limits of what can run in a browser without too much lag. However, if you want a browser version to play on a Chromebook, you can actually get the linux version running on chromebooks with linux beta.
But how can I play on a school Chromebook? is there such as paying ingame and going to a site with the file and you have to log in to check if you have paid? then you can play
There are a bunch of videos showing how to install steam on chromebooks - if you get steam installed, it's easy to run Aground from your steam library (and the itch.io purchase comes with a steam key).
The easiest way is to actually install steam - download the steam linux deb, and then right click and install with linux beta - it'll handle all the commands necessary and has a GUI so you don't have to deal with the terminal.