This was my spontaneous answer to the part of the question saying that "different places in the literature the same name is used for two different mathematical objects." Plus, I am thinking of a notation as a written form of a name.

Wow, amazingly the "three options" you have mentioned is more or less the conclusion of my research as well, though I have used quite a different language. I'll send my papers to you, the published one, that is more about the history of equivalence as a relation and the under review one that is more about equivalence classes. It is in this second paper that I nearly came to the conclusion that you have mentioned.

@TomCopeland That is not true at all. Remeber, the pain Frege had to define "direction". Basically, he wrote "The foundations of arithmetic; a logico-mathematical enquiry into the concept of number" to justify the legitimacy of definition by abstraction. Could you please consider what happened at "the moment someone took in the notion of", say, direction?

Dear Peter, please let me read and digest your current answer before adding anything to it for a while :)

Please read this comment before the second comment :) But then, there is your second point that basically implies there is no first equivalence since it is "as old as human thought or at least human use of language". With this point, I strongly disagree. Consider that in all your examples, you have first abstracted a concept, formed it, named it, and then saw the equivalence of the objects under it. In the definition by abstraction, you are moving in the opposite direction.

As something that "is not intended as an argument by authority" :) let me quote Hermann Weyl: "In looking at a flower I can mentally isolate the abstract feature of colour as such. This act of abstraction would here be primary while the statement that two flowers have the same colour 'red' would be based on it; whereas in mathematical abstraction, it is the equality which is primary, while the feature with regard to which there is equality comes second and is derived from the equality relation.

Let me separate two points of your answer. The first point is that I have assumed that the development and acceptance of "definition by abstraction" has a close tie with development and acceptance of "the abstract method". I cannot comment on this point more since it was the result of my failure to attribute the "first" equivalence to any of the candidates I had: Euclid, Gauss, Dedekind, Pasch, Cantor, Frege, and Russell.

Your answer has encouraged me to have a deeper look at Essays on the Foundations of Mathematics by Moritz Pasch since Pasch has played a recognizable role on what he calls "Implicit Definition" (Definition by abstraction?). As soon as I move to another point made in your answer (there are too many) I let you know :)

I just know that story only up to the chapter that people had difficulty grasping 1 as a number. Strangely, I never read the next chapter. However, I see how it is related to my question now upon your mentioning.