Scientists also write; writing isn't the point, the point is logically consistent arguments (or arguments for a particular logical system), which philosophers attempt to make. "Mathematician", "programmer", "logician" and in many cases "engineer" are also self-appointed titles, yet in both cases we must say that a mathematician and engineer's work could reasonably be said to be confined to writing and designing. The fact of the matter is that logical claims do not require "proof", but the relevance of theories to the world as observed by science does require support from the scientific method. This is why philosophers strive for their systems to be consistent with the current state of science (whether physics, biology or sociology, etc.)
>And as you correctly say: this concept predates science and enlightenment.
It not only predates, it is ontologically prior, meaning that for science to be carried out, there must be a justified philosophy of science for it to have any basis. For instance, we might ask, "why falsification?" - a question that, if unanswered, renders science without a logical basis. You may reply that science is useful whether or not we have a logical basis, but usefulness in turn requires some justification; after all, many things are useful but we do not carry them out for other reasons.
>In the case of traditional philosophy of science there is even a contradiction in terms: the statements are dogmas that defy falsification
This is not a contradiction in terms, since most philosophers of science are not pure falsificationists. Falsification is not the only way to gain knowledge generally (logical arguments and mathematics are unfalsifiable; even the statement "science must be falsifiable" is unfalsifiable), but after Popper there are very few pure falsificationists in the philosophy of science, for various good reasons[0].
>So according to the own criteria of the philosophy of science of that time they do not differ from religion.
Philosophy of science is very much the equivalent of atheism, or agnosticism when we're talking about dogmas such as falsificationism or positivism or any other popular way of doing science that is unjustified.