It is hard to say what "official" means exactly, it is not like there was a bureau of terminological standards. But "real numbers", "real values" and "real quantities" were certainly widely used long before Dedekind, and the erasure of the distinction between rationals and irrationals where it is not relevant is even older, it goes back to Stevin (1585), see How were irrational numbers accepted by mathematicians? It was supported by Stevin's normalization of arbitrary decimals and the idea of generation of "quantities" by continuous motion, upon which Newton, Euler and Cauchy founded calculus. Both served as an early substitute for the arithmetical constructions of real numbers a la Bolzano, Meray, Dedekind, Cantor, etc.
Descartes did not employ "real numbers" as a term, but he did start the tradition of inserting the adjective to distinguish them from imaginary numbers. Here is the relevant quote from La Geometrie (1637):
"For the rest, neither the false nor the true roots are always real, sometimes they are only imaginary, that is to say one may imagine as many as I said in each equation, but sometimes there exists no quantity corresponding to those one imagines."
Since by "roots" he means intersections of arbitrary polynomial curves with the axis he clearly did not distinguish rational from irrational. Newton, Leibniz, Euler and others continued in the same vein, with "real" becoming a term. Leibniz's unpublished 1675 paper (see A Contribution of Leibniz to the History of Complex Numbers by McClennon) discusses the cubic as follows:
"But if now a simple, that is, a linear equation, is multiplied by a quadratic, a cubic equation will result, which will have three real roots if the quadratic is possible, or two imaginary roots and only one real one if the quadratic is impossible"; "How can it be that a real quantity, a root of the proposed equation, is expressed by the intervention of the imaginary?"
Euler's Introductio in analysin infinitorum (1748) uses "real numbers", "real values" and so on routinely. In chapter I (Bruce's translation) we read, for example:
"Thus, even if this function $\sqrt{(9-zz)}$ at no time is able to receive real numbers substituted in place of $z$ greater than the number three, yet imaginary values of $z$ may be attributed"; "The three values of $Z$ for that one corresponding value of $z$ either are all real or one will be real, while the two remaining are imaginary."
Considering the general influence of Euler's textbooks on the subsequent notation and terminology his use can be said to make it "official". By the way, when Euler uses "numbers" without qualification he means integers, not real numbers, see e.g. chapter XVI.
Cauchy's Cours d'Analyse (1821), another influential textbook, adds "real" in the same context, but Cauchy typically talks about (variable) "quantities" or "functions" and their values, not "numbers". In section 7.1 (Bradley-Sandifer's translation) he writes:
"In general, we call an imaginary expression any symbolic expression of the form $\alpha+\beta\sqrt{-1}$, where $\alpha$ and $\beta$ denote real quantities"; "Given this, any imaginary equation is just the symbolic representation of two equations involving real quantities".
In Note III he even uses "real" without direct relation to the imaginary, in the intermediate value theorem:
"If the two quantities $f(x_0)$ and $f(X)$ have opposite signs we can satisfy the equation $f(x)=0$ with one or several real values of $x$ contained between $x_0$ and $X$".
