It is better than "Incognito mode".
> "You’re browsing as a Guest"
> "Pages you view in this window won’t appear in the browser history and they won’t leave other traces, like cookies, on the computer after you close all open Guest windows. Any files you download will be preserved, however."
It's kind of like Incognito, except none of your preferences or extensions are there, either, it's just an entirely new profile that self-destructs when you close it.
The OP's detector considers a guest profile not to be Incognito mode.
Re your question: I use Firefox with 1st party cookies only (and the other associated privacy options) and it works pretty good. Some WebApps break, but very rarely.