This is a problem with copyright law. There is no way for an end user to determine the copyright status of anything on the Internet, you can only make an educated guess.

It's pretty simple in the US. A work has a copyright regardless of whether it's registered or a notice placed on the work. Registration provides easier means of asserting your copyright but you have a copyright as soon as you create the work. If I wrote a handwritten note about OpenAI on a cocktail napkin then I have copyright over that work barring some challenge to whether it's a "creative work" or not. It doesn't matter what the medium is, or how it's shared. The internet makes this challenging in that it's essentially a shared technical means of disseminating the work, but the work remains copyrighted no matter how publicly available it might or might not be. It's just a matter of the rights-holder asserting their right. Which is something NYT does with their paywall all the time.

As read you are asserting that public domain doesn't exist.

Not only that, if something is available on the Internet, and still under copyright, you have no way of knowing whether the website is authorized to distribute it or not.