It seems that the term "Diophantine equation" has been around at least since the second half of the 19th century, since the historian Hermann Hankel writes (polemically) in the chapter on Diophantus in his Zur Geschichte der Mathematik in Alterthum und Mittelalter (p. 163):
At this point, there is a mistake to be corrected, a mistake that is reinforced by false nomenclature and therefore, or so I fear, impossible to weed out. In education, one designates linear equations of the form $ax+by=c$, that are to be solved in integers $x,y$, as Diophantine. Now, not only did Diophantus not know the solution method of these equations, that in the West was first obtained by his commentator Bachet, but the very problem would have been utterly alien to him, since he never fixes the condition that his solutions should be integral, but is completely satisfied with rational solutions.
(Italics are mine.) Now, in connection with this passage, I have the following questions:
- When did people start talking/writing about "Diophantine equations"?
- Were Diophantine equations originally considered as "polynomial equations to be solved in rational numbers", in accordance with Diophantus' own preference, or was the mistake that Hankel aims to correct made from the beginning?