Learn English – A collective term describing both pick-up and drop-off

I once knew a fellow who was prone to this specific fault. He wasn't a "circumlocutor," as that would mean a person who "talks around" a topic without coming to the point. Rather, this person would trail off into arabesques of detail that were unneeded (and unwanted). Let me give you an example:

"I had a teacher who wore shoes like that - he was from Amherst, you know, the school in Massachusetts, and I got a speeding ticket there from a cop who had a radar gun that looked like the things you'd see in an RKO serial, like Flash Gordon, and y'know I think we ought to have those again in theaters except that theaters today just show trailers, and do you know why those things are called "trailers" anyway? They don't "trail" anything, they lead it off so maybe they should be called "headers" except that a header is maybe something you'd see in soccer, which is called "football" in most countries where it's played and isn't that weird?"

And in this, I believe that English suffers from a fault in that we do not have a precise word for a person who talks this way. We could coin "digressor," perhaps.

The word twilight actually means what you're looking for:

twilight, n.: the light from the sky between full night and sunrise or between sunset and full night

Animals that are active during twilight are said to be crepuscular, so if you're looking for an adjective to apply to that period, crepuscular (which is itself derived from the Latin for twilight) should serve.

Edit: Lacking the context in which you wish to use the term you're looking for, I can offer these suggestions:

Sunset/sunrise occur at the instant that the sun's upper limb is tangent to the horizon. You could then, for example, have science officer of some enterprising starship reporting, "Two degrees to limb tangency, Captain" to mark some impending sunrise/sunset/moonrise/moonset/et ceteraset.

Sunset/sunrise occur at a precise zenith distance of 90.8333 degrees. You could coin a term for this value, so that the same science officer could report, "Two arc seconds from zenith distance alpha, Captain."