Translation:We don't like cloudy days.
I disagree entirely, Duolingo translates how it wants to and follows no particular method. Sometimes they want word for word literal translations and other times they just make stuff up that is more vague and misses a number of words that are in the Chinese statement itself. This holds true in all of the languages I am studying here, but Mandarin Chinese is the worst I've come across. It's a guessing game.
The only way I can beat a level like this series on the weather is to write down every answer in Mandarin characters, pinyin, and English, so I can go back to refer to the answers they want, and that usually is every translated answer to each question at the first series of questions. In this particular weather lesson, I've written down five pages of answers. How on earth are you supposed to remember all of those exactly without writing them down? And don't write too many original answers, because if you do you'll find yourself being marked wrong for doing simple things like writing "do not" instead of "didn't" in some cases.
This is a constant curse with this course, mainly because they are doing zero maintenance on adding an appropriate set of answers to most questions, whether you report them or not.
"We do not like cloudy skies." should be accepted as well based on the definition at https://en.wiktionary.org/wiki/%E9%99%B0%E5%A4%A9#Chinese and that "天" can mean "sky" as well as "day".
Even based on the hints, 阴天 means 'cloudy', which makes the literal translation go 'we don't like cloudy'. I can't see how the 天 would mean particularly 'day(s)', so 'we don't like cloudy weather' or even 'we don't like cloudy' should be acceptable.
It feels to me that DL more than would be good omits the, e.g., 天 with weather, or 色 with colours. For example, et DL I see sentences like 我喜欢这个红椅子, while my Chinese teacher (a native) will say 我喜欢这个红色的椅子. Can anyone explain the usage of these?
As for the difference in your 2 sentences, I think your teacher's is more "precise" and DL's is more "casual". But as for "we don't like cloudy" - no native speaker would say just that. They might say "we don't like WHEN IT'S cloudy" or "we don't like cloudy days, weather, skys etc."