The browser version doesn't have the start where the whole situation gets explained. The Windows version has this.
The true ending also doesn't seem to work in the browser version; I tried twice.
Same as the second and third dream; Both are missing on the browser version, but present on the Windows version.
Same as dead end number 7.
Very cool game though.
Very interesting, this might be because of how your browser handles cookies? RPG Maker saves the global variables (aka all the data that's responsible for the things you mentioned) via cookies in your browser.
Maybe trying out a new browser (or private mode) might help with that? (That's just a guess though)