I’m defining the division operation, not the quotient

Yep, the quotient is the result of Division. It's right there in the definition in Euler. Dividend / Divisor = Quotient <= no reference to multiplication anywhere

Yes, the quotient is obtained by dividing… Now define dividing.

You not able to read the direct quote from Euler defining Division? Doesn't mention Multiplication at all.

The actual is the one I gave

No, you gave an alternative (and also you gave no citation for it anyway - just something you made up by the look of it). The actual definition is in Euler.

That’s why I said they are also defined based on a multiplication

Again, emphasis on "alternative", not actual.

implying the non-alternative one (understand, the actual one) was the one I gave

The one you gave bears no resemblance at all to what is in Euler, nor was given with a citation.

Feel free to send your entire Euler document rather than screenshotting the one part

The name of the PDF is in the top-left. Not too observant I see

you thought makes you right

That's the one and only actual definition of Division. Not sure what you think is in the rest of the book, but he doesn't spend the whole time talking about Division, but feel free to go ahead and download the whole thing and read it from cover to cover to be sure! 😂

Note, by the way, that Euler isn’t the only mathematician who contributed to the modern definitions in algebra and arithmetics.

And none of the definitions you have given have come from a Mathematician. Saying "most professions", and the lack of a citation, was a dead giveaway! 😂