I was in dark mode at the time of testing, and didn't realise there was any different at all until you mentioned 🙂
I'd agree in light mode there is likely enough distinction with the dark address bar. Perhaps something in the URL bar (to the left of the padlock) like an Orion icon with a pair of sunglasses on? I'm definitely not a design language person though... I just think we need a more visible cue in the UI for new users - even with colour, it's hard to "learn" which colour is normal/right. For dark mode, I think an icon or similar might be needed.
In dark mode, an unfocused non-private window is pretty much visually indistinguishable from an unfocused private window, as you lose the purple-ish tint.
Any UX gurus got any ideas?