Yes. There is the difference what environment to be mentioned. However, I think the difference is already almost listed up in the definition you gave.
野外{やがい} seems corresponding really well between English and Japanese.
But, "suburbs" is bit odd to me. If you are in 「野外{やがい}」, there should be few buildings around you. So, the word is not really relevant to residential area.
「野外{やがい}フェスティバル」is held normally nearby mountains or parks. However, a rock festival called "Summer Sonic" is normally held in "Marine Stadium in Chiba" which is not far away from business district. But, the stadium is open-air. So, the festival should be categorized in 野外{やがい}フェスティバル.
屋外{おくがい} is used to describe for "outside the building". So, the word only "outdoors" is bit odd to me. The dictionary probably wants to say things like 屋外広告{おくがいこうこく} : Out of home/ Outdoor advertising.
In English, the word "outdoors" seems just outside the building. But I believe it is often used to describe for hiking area where there are few buildings.
戸外{こがい}, a similar word, is used to describe outside the residential house.
The picture below is an explanation I have found and I believe this is a standard interpretation.
Reference and source of the picture : 野外と屋外と戸外の違いとは
