No, you are not correct. How many people have to tell you this?

It absolutely works the way I said it does, because I have seen it work that way. Just because you accidentally turned off iCloud Photos in your Apple Account settings on that Mac (or some other similar issue) does not mean it does not work this way when properly signed in.

If you want something to try, go to System Settings -> Apple Account -> Photos and see if "Sync this mac" is turned off. It needs to be on. There could be other ways that this feature is disabled, but that is one of them.

Not seeing something work is not evidence that it does not work. You have not seen it work, but that is not proof it does not work.

Seeing it work is evidence that it works. I have seen it work.

Other people have seen it work that way, and their replies are all over this thread. Apple documents that it works this way.

Yes, it will upload photos to iCloud if enabled, but it also downloads them.

When you take a new photo, it synchronises with all your devices, and therefore you see it on your Mac, iPhone, etc. However, if you get a new Mac (I got one because my library was under capacity), Photos will not start synchronising your 10-year-old photos until you process them through the phone.

I hope I've made it clear now.