Chrome Incognito and FireFox's Private Browsing are functionally identical. The caveat that the author highlights is how the Internet works. Of course sites have a record of your visit... they have to, to feed you the page! The disclaimer is to make sure that people know Incognito mode is like wearing an Anonymous mask, not like being invisible. And if you go up to an ATM dressed like V, but get money out of your credit card, then obviously the bank knows who visited the ATM despite the mask.
This basic ignorance of how cookies work is pervasive.
Private browsing opens your browser in a blank-slate mode. Generally, no plugins, no cookies. That means Amazon doesn't know who you are, so you can't one-click buy. Your news-reader makes you log in again. It takes longer to access your email because Gmail makes you log in and re-affirm your authenticator. Your ad blocker is disabled. Your CSS fixing plugin is blocked.
This is not how I want to use my computer, logging in to every single site every single time I visit despite being on a trusted device. We have plugins and cookies for a reason, because they make the Internet a more useful tool. They also have nefarious uses, but saying that the Internet should throw out all convenience to maximize security is ignorant of the reality that people will just switch to the more convenient browser.
What we need is not a better incognito mode, but for tech journalists to stop pontificating about technology they do not understand.
If you really want to improve your anonymity online there are plugins that allow you to whitelist 'safe' cookies, and trash or block all the others. That plus plugins to block third-party widgets allow you to get 99% of the functionality from the Internet with only 1% of the spying. But these plugins take work on your part, to identify what sites and cookies you trust. Most people are too lazy. And the browser has no way of knowing for you. For example, I may want Amazon to remember me so I can buy with one click... you may not because you don't trust Amazon's tracking of what products you look at. The browser shouldn't be deciding that for you, but making choices like that for every site is a pain few users will bother with.