@FrancoisZiegler Thanks very much for the time and effort you put on this question. It was indeed an amazing journey for me.

@MichaelGreinecker Shall we credit König also for the term "equivalence class"? Or it is just "equivalence relation"? Could you give a reference for the exact paper of his that you saw the term? What is the date?

The meaning that I want is exactly the one we use today. It is for sure that I cannot read those papers. I now understand why a wise man once said "knowing three languages is better than two"!

I wait a few days to see if there is an earlier date. If not, I "accept" your answer. Considering that you reduced the time suggested by Henry Cohn in a few minutes, I am sure a few days would be fair to history.

How did you find that? Once upon a time, before the age of MO, I did a two years search and I failed. Shame on me :)

@HenryCohn That's great. The time gap is now only 24 years.

@PeterMichor He (!) indeed defined "equivalence relation" and "equivalence class" in "Theory of sets". But, there is a big time gap between what I quoted from Russell and the Bourbaki's book. It is very strange that such an important notion remains unnamed for such a long time.

@ToddTrimble Dear Todd. I am not sure it is a good idea since it may not reflect the previous answers. It won't be fair to the people who answered the old question. Having said so, I leave the rolling back decision to you

more importantly, suggests a complete different viewpoint (from Schoenfeld and alike) that I think somehow shows the framework you have mentioned is not working for all. I am very interested in continuing this discussion. But MO is not a good fit for that. Perhaps we may keep in touch by e-mail.

@BenjamiDickman Dear Benjamin. Thank you for the answer. It is now for years that I am teaching a course called "theories of problem solving", an important part of which is reading Schoenfeld' book and his articles and also other people's article. Having said so, I really appreciate you have mentioned that literature since many in MO do not tend to read it. It was why I tried to use a jargon-free language to express my question. In particular I referred to Hadamard' work (The Psychology of Invention in the Mathematical Field) since he uses a language more familiar for MO viewers and