That lack of delineation is also an issue, but a separate one. That said, I’d think an average user would think doing a Google search from an incog tab would be anonymised and not tied to them because of the privacy incog grants (or more accurately, doesn’t). There’s reasonable arguments to be made on either side of this point, but I think that Google have been intentionally misleading - which is now creating problems for them, motivating this change.
Again, all the information Google present when opening an incog tab would lead someone to the conclusion that Google won’t track them. Unless I’m mistaken, when this came up years back, Google explicitly denied tracking people in incognito mode, and they’re only changing their disclaimers now in response to a multi-billion dollar lawsuit.
If Google specifically denied tracking that’s definitely misleading, but I’m unable to find a source for it and don’t recall it myself.
Saying that the sites you visit track you would absolutely lead me to believe that search engines sites are included. Since it would not be possible to provide results for the search without knowing what was searched for by the user. And where would they send those results to without knowing the users IP or other form of network address? It just doesn’t make any sense to think a search engine would not know who searched for what, since it is required for them to function.
You don’t visit the site when you punch a query into your browser search bar.
Ultimately, Google are making the change they are because they know how deceptive they were being. Google knows it, I know it, Google seems concerned the courts know it, I’m not sure why you’d choose to dig in on this one.
