The server sees your IP regardless of your browser being in private mode or not. And yes if they have the IP in the logs and have them accessible to query, they could easily correlate all requests from the same IP and determine what they were accessing inside private windows or not.
There's a gulf between the "I don't want companies tracking me" use case of the OP and the "If I get caught spying I'll be executed" use case that Tor addresses.
For the uninitiated, don’t use tor to solve the second goal here without doing proper research. Just using tor is not effective against nation state actors:
Yes, I would hope that anyone whose freedom or life is at risk isn't depending on Tor to keep them safe without knowing what they're doing.
In fact, I'd go one step further and recommend that if your freedom or life is at risk, then don't use computers or phones to do things that states might execute or imprison you for at all.
That gulf is readily filled with a garden-variety VPN, which works if you're doing the kind of thing where you don't want the web site keeping tabs on you but figure the VPN's reputation is enough to keep them logging your connection and doing something nefarious with that information.
