what is the difference between reverse, dark and leakage current of a (photo-)diode? From my current understanding it all seems to be the same?
What's common for all these currents is that they are reverse, i.e., they flow when a reverse voltage is applied to a diode, but there are some differences.
A dark reverse current in a photodiode flows when there is no light and is, naturally, different (smaller) than the reverse current in a photodiode exposed to light.
Although some regular diodes could have some sensitivity to light, the term "dark current" is not used to describe their reverse current, since it would not be much different from a "non-dark" reverse current. Instead, we use the term "reverse leakage current".
With that said, reverse dark current and reverse leakage current have similar underlying mechanisms. They both increase with the temperature and the reverse voltage and, in most cases, are both considered undesirable.