Downrange is the horizontal distance traveled by a spacecraft, or the spacecraft's horizontal distance from the launch site.
Spacecraft don't travel horizontally. I don't even know how the word "horizontal" and the word "spacecraft" can exist in the same sentence. Maybe the word "projected" would be helpful here?
This is simply the distance across the ground from the launch site.
This is the accepted and highly upvoted answer. If I had to write an equation from this explanation, or interpret what 500km downrange means, I'd be hard pressed. It's a great answer to get the general idea, but a bit wanting from a mathematical or orbital-mechanical perspective.
Is there an official or generally accepted, precise definition for how one would calculate downrange distance? I can imagine if I have an orbital plane, then downrange and altitude might be referenced to a sphere or an ellipsoid (Earth surface model) and thereby
could be could have been used to define a position fairly precisely. Does such a definition exist?
To better illustrate why the question is not trivial, let's abstract the mathematics out of history temporarily. Imagine you would like to, or have been asked to calculate a down range distance. What is the first question you might ask yourself:
"If I have ECI coordinates of a launch site and a spacecraft, how would I calculate the downrange distance correctly? What model should I use for the Earth's surface? Or should I project along a sphere with the same altitude as the launch site?"
I'm looking for an authoritative answer, not what it probably means or could or might mean, how to "think about it as...", or "it doesn't matter" — it has certainly been relevant historically. Thanks!
EDIT: To elaborate on the previous sentence defining the scope of the question, a look at NASA's 254 page Apollo 11 Press Kit will show a dozen numerical downrange values with single digit nautical mile precision, (and one with decimal precision). To get a large downrange value to 1 nautical mile precision, one needs to choose a specific model for the shape of the Earth, at least sphere or ellipse.