Progress is transferrable, but other than that, you're correct.
Downloaded Itch and Steam versions keep their saves in the same folder, so if you've played the download, it'll all just work. If you've played on browser, you'd have to export/import. The only transfer which doesn't work is you can't go from a newer version to an older one. So once you play on Steam/paid Itch, you won't be able to load that save into the browser or free Itch versions.