In fact, it's a rule of thumb that the higher you go the less time you spend in Earth's shadow: the shadow not only gets narrower, your orbit gets bigger.
Rules of thumb are never intended to be considered true in all cases; we use them in a pinch or when we plan to go back later and calculate rigorously (and often never do).
I'm wondering how often (if ever) this rule of thumb fails; if there's a class of circular Earth orbits where it's ever, or even frequently not true.
What's left out of the explanation is that when the satellite's orbit gets bigger, the satellite also moves more slowly.
So while the fraction of the orbit that's spent in eclipse is very likely to decrease with increasing semi-major axis, it's not clear that the absolute duration of a given eclipse will always decrease with increasing semi-major axis as well.
So I'd like to ask:
Question: Are there any Earth orbits where the duration of eclipse increases with increasing semi-major axis with all other parameters fixed?