Translation of Dr. rer. nat. outside of Germany Psychology at universities in Germany is either in the natural sciences or in philosophy, depending on the university. This affects the kind of academic title you get. People doing their doctoral thesis either get the doctor in natural sciences (Dr. rer. nat.) or the doctor in philosophy (Dr. phil.).

If I am not mistaken, most doctors in psychology (internationally) are Ph.D.s. So my question is, how would you refer to your title internationally? As a doctor of natural sciences? As a Ph.D. (even if you have a Dr. rer. nat.)? Are there legal requirements? I am thinking about a website that is generally understood internationally and -- perhaps -- about business cards.

(And yup, personally, I think -- and act if -- psychology is closer to natural sciences than philosophy.)

(I'm sorry if I can't specify the question more closely. There's a similar question (Choosing a title to hold upon completion of a doctoral degree: “Dr.” vs. “Ph.D.” [duplicate]) but I don't think it really covers this issue.)
In the US, most doctorates in science and engineering are also a PhD even though our degrees aren't in philosophy. While other degrees exist, they are less common. I would suggest translating your degree as a PhD since most international audiences will understand that you mean an academic doctorate. 

That being said, I was able to get what Dr. rer. nat. meant, too, so you might decide to put that and not worry about it. Most people have the ability to search the internet when they are confused.