Due to the geometric formulation, Kepler's Astronomia Nova is full of elaborate illustrations of ellipses, epicircles and whatnot. It is not necessary to spell out the position of points in Cartesian coordinates, because the relation between points are specified in angles, lengths, and geometric constructions.
Kepler presents the use of ellipses for planetary motion in Astronomia Nova (1609, originally in Latin; translated by William H. Donahue, 1992, Cambridge University Press), when René Descartes was thirteen years old. Paging through introductory words by Kepler himself, translators, and commenters (Max Caspar, Kepler, 1993, Dover Publications) is highly recommended because, oh, do they throw shade.
The definition of an ellipse as a conic section, as known to the Greeks, already is a formal definition. Kepler actually cites the Greek philosophers for many propositions. For an in depth historical view, see chapter 2.1 of The Ellipse: A Historical and Mathematical Journey by Arthur Mazer (2010, Wiley).