Learn English – Generic term for songs, movies, TV episodes, etc

I'm afraid that there isn't a single word meaning our company as opposed to another company in English. If it's too confusing to use just plain company, then you may have to come up with some form of us, this company, our company, etc. on the one hand, and affiliate or associate companies (or client, if that's applicable) for the companies "our company" is working with.

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."