Classic for Europeans. As an American East-Coaster I'd consider the White-Tailed Deer an example of a classic deer.
While yes, the white tailed deer is the normal american deer, i wouldnt call it a classic animal.
Correct me if im wrong, but Afaik the white tailed deers impact for american culture can be boilee down to wild game.
The red deer on the other hand is a culturally loaded animal, seen as the standard for things like fairytail and the deer that you would see on things like crests and artwok.
The red deer is deeply integrated in european and with that also western culture in general to some extend, aswell as just being one if europes few standout species in general.
Over here, the Red deer stands on even ground with the wolf, fox, white stork, eagle, lion and bear as one of the most iconic and most culturally important animals, so really calling the majesty of the forrest anything but the classic deer would be a disservice from both a historical and cultural view.
If you see a deer in pop culture, very high chances that its a red one, for example in the crest of the golden deer in fire emblem three houses.
The only cultural reference for the white tailed deer i could find was that in the movies, bambi was made one, as americans wouldnt be familiar with a european roe deer, bambis original origin.