Translation:We don't like cloudy days.
That's actually a better translation. Since 天 translates to both day and weather, among other things. You should definitely report it :)
There is no word for weather 天气 in this sentence. Duolingo usually wants us to translate word for word more than just the meaning.
"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."
"We don't like overcast days" would be idiomatic and one might even get away with eliding "days" on the perhaps shaky theory that "overcast" could be a noun as well as an adjective.