After saving the remixed project, the URL in the browser is not updated. So you are not aware which project is loaded in the editor and, if you reload, the original project is loaded not your private copy. If the project is saved as "shared", URL Is modified as expected.
Hmm, so, we should change the URL to something that doesn't look like a shared project, when you change anything in the project you just loaded?
Most editors have a visual indicator of unsaved changes...
But it will be enough to change the URL when saved to the cloud.