I am teaching College Algebra and Pre-Calculus this year out of two college textbooks, Ratti & McWaters for the former and Blitzer for the latter. I happened to be teaching conic sections in both classes on the same day. Here’s the Pre-Calculus take on “parabola”:
A parabola is the set of all points in a plane that are equidistant from a fixed line, the directrix, and a fixed point, the focus, that is not on the line.
Simple, clear, exact.
And here is College Algebra (mind you, the “easier” course meant for those not necessarily taking any higher math):
Let l be a line and F a point not on the line l. Let P be the plane determined by F and l. Then the set of all points P in the plane that are the same distance from F as they are from the line l is called a parabola. Thus, a parabola is the set of points P for which d(F,P) = d(P,l), where d(P,l) denotes the distance between P and l.