philosophy is a part within the framework of religious order

I know that this is not the best kind of response, but as a philosopher I must come clean that I am not always 100% what we mean with philosophy. Or in other words, there are so many understandings of philosophy out there.

In my view philosophy was not taken over by religion

True, but I think, if I understand Pierre Hadot's argument correctly, the split between religion and philosophy in ancient Greek was different to our understanding of philosophy vs. religion today. But I totally understand what you are saying as well. (If I remember correctly, and correct me if I am wrong, but Socrates believed in the gods but different to those of ancient Greece. Today, philosophy is sometimes synonymous with a-religiosity.)

Those, who were and are particularly good at making sense or asking surprising/challenging questions became known philosophers

Again, according to Pierre Hadot, sorry for bringing him up the whole time, makes this exact split. To him, Hellenistic philosophers and those before Hellenistic philosophy, did not make this split between the text and the exegesis of a text, as you made. Or, there were only philosophers living their philosophy, there was no exegesis as such or as we understand it today. There were only philosopher living their philosophy, and most of them did not even write this down. If you could not practice your philosophy, that is, live your philosophy, it was not a worthy philosophy. But since then, the image of philosophy changed, and I personally think just living one's philosophy today is not all that viable. (But this is maybe an argument for another day.)

Since I don't know myself a unified philosophy but many schools and streams of philosophy.

Exactly, I totally agree with you on this. Today, philosophy is radically different to philosophy (again, according to Hadot) practised in ancient Greece.

specific terms or phrases

The easiest that comes to mind are some of the phrases from ancient Greek Pyrrhonism. They had a phrase ou mallon, I think that is the spelling, but essentially it means "no more this than that". In my own life, I try not to add additional beliefs to things, for example, that it is inherently bad/good to be hot or cold, as per the Pyrrhonians, this extra "layer" of beliefs is what causes disturbances in life. Maybe this makes sense? I can give other examples as well.

Buddhist teachings it is mentioned, for example, that between the master and his disciple the master answers in secular terms when his student asks a metaphysical question and if he asks an everyday question he answers in metaphysical terms.

I really like this example, thank you so much! In the back of my mind, very deeply buried away, I am familiar with this example, having read about it many moons ago.

Thanks again for this thoughtful reply! I hope that you find the answers here engaging with your reply.