Origin of the term "Diophantine equation" 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"? 


And also: 


*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?

 A: Google Books provides partial answers to these questions.
If we allow ourselves the freedom to consider not just the phrase "Diophantine equation" but also the phrase "Diophantine problem" then it was used much earlier than the late 19th century.  For example, An Introduction to Algebra by Bonnycastle and Maynard (1788) says that "Diophantine problems are those which relate to the finding of square and cube numbers, &c."  In other words, Diophantine problems are problems like the ones Diophantus studied.  It was apparently not yet standard to generalize to arbitrary polynomial equations, let alone settle on the rational/integer question.
In the Edinburgh Review (1818) it says, "On the subject of Indeterminate Problems, we must remark, that the Indian algebraists had gone much further than this, and had resolved those of the 2d degree, or such as are properly called Diophantine, where the question is how to render a certain quantity rational, either in fractions or in integers."  So here is an explicit allowance for non-integer solutions, though still not explicitly generalizing to arbitrary polynomial equations or using the magic word "equation."
If we insist on the exact phrase "Diophantine equation" then there are numerous 19th century occurrences, most of which simply use the term without giving a formal definition.  The Calcutta Review (1844) contains a sentence of the form, "Now it is evident that this is nothing else than the Diophantine equation … in which a is a known number and is required to find x and y in whole numbers."
As they say on Facebook, it's complicated.
A: There is a website, Earliest Known Uses of Some of the Words of Mathematics. Some entries are, 
DIOPHANTINE ANALYSIS (named for Diophantus of Alexandria) occurs in French in a letter of March 1770 from Euler to Lagrange: “ce problème me paraissait d'une nature singulière et surpassait même les règles connues de l'analyse de Diophante” (“this problem appeared to me to be of a singular nature and surpassed the known rules of Diophantine analysis”).
Lagrange used “analyse de Diophante” in a letter to D’Alembert in June 1771.
“Diophantine Analysis” occurs in English in the chapter title “Demonstration of a Theorem in the Diophantine Analysis. By Mr. P. Barlow, of the Royal Military Academy, Woolwich.” in The Mathematical Repository, New Series, Volume III (1809) page 70.
[This entry was contributed by James A. Landau.]
DIOPHANTINE EQUATION. Felix Klein used Diophantische Gleichungen in “Die Eindeutigen automorphen Formen vom Geschlechte Null” in the 1892 issue of Nachrichten (page 286): “Die Relationen kann man in Diophantische Gleichungen umsetzen, welche dann leicht übersehen lassen, unter welchen Umständen Multiplicatorsysteme möglich sind, und in welcher Anzahl.” [James A. Landau]
Diophantine equation appears in English in 1893 in Eliakim Hastings Moore (1862-1932), "A Doubly-Infinite System of Simple Groups," Bulletin of the New York Mathematical Society, vol. III, pp. 73-78, October 13, 1893 [Julio González Cabillón].
Henry B. Fine writes in The Number System of Algebra (1902):
The designation "Diophantine equations," commonly applied to indeterminate equations of the first degree when investigated for integral solutions, is a striking misnomer. Diophantus nowhere considers such equations, and, on the other hand, allows fractional solutions of indeterminate equations of the second degree.
DIOPHANTINE PROBLEM. The phrase "Diophantus Problemes" appears in 1670 [James A. Landau].
The OED2 has the citation in 1700: Gregory, Collect. (Oxf. Hist. Soc.) I. 321: "The resolution of the indetermined arithmetical or Diophantine problems."
