Thanks, Vincent. I do not really use other browsers. I just tried MS Edge that I have here and it did not display anything at all. As I said I always have to use save to file or save to string in other twine games. I played with cookie settings and that did not do anything, besides the saved data appears in the Local Storage rather than cookies in the Firefox storage inspector, which are separate entities somehow (see attached). Anyway, I think I'll give up on this for now.
PS Apparently localStorage is HTML5 equivalent of cookies:
You must be registered to see the links
and it looks like in Firefox localStorage follows cookie privacy settings:
You must be registered to see the links
so it should work if the cookie settings are permissive, which I tried...
Edit: for anyone else with problem you can output the localStorage into string in development console with this:
JSON.stringify(window.localStorage)
save the text for later and next time recover with:
tmp=JSON.parse(saved text heer);
for (var key in tmp) window.localStorage[key]=tmp[key];
you need to click on any link on the page for the "load game" to show up.