My understanding was always that Incognito mode was only intended for instances when you didn't want browsing history saved, not instances where you wanted to protect your privacy. That said, I think an 'enhanced Incognito mode' which does protect your privacy that's built-in could definitely be a good addition.
The point of Private Mode is also to start with a blank session, without any cookies.
Service providers, such as Google or Facebook, are tracking users even when they aren't logged in, by setting unique identifiers in cookies with a very long age. Then as soon as you login, they can even correlate those identifiers and their history to your name should they want to.
Facebook for example is known to build profiles on people that do not have a Facebook account. And given that many websites are integrating with services provided by such companies (e.g. Google Analytics, Facebook buttons are everywhere), it's not like you have to go to Google.com or Facebook.com to be tracked.
So that's the point of Private Mode, because in Private Mode all they have is your IP. And in case we're talking about IPv4, we could be talking about a home connection, or a public Wifi, or a work connection, so to track users one needs to take a look at usage patterns coming from the same IPv4 and make a decision - home connections are what you want, as otherwise too many people are connecting from the same public wifi or work connectin.
And given the shortage of IPv4 addresses, ISPs have switched to dynamically allocated IPs at least for home subscriptions. Mobile phone operators are doing the same thing - an IPv4 coming from a mobile phone doesn't even tell you the user's city.
It will be interesting to see how we'll be able to protect ourselves along with the switch to IPv6, but in the meantime, yes Private Mode has everything to do with privacy.
