Generally orange diamonds are called diamonds.
There is a wide range of descriptors for diamonds that contains orange, such as "...Reddish-Orange, orange, Yellowish-Orange, Yellow-Orange, Orange-Yellow, Orangish-Yellow..."
Then in brown diamonds, you can find ". . .Cognac is usually used to describe a diamond that is Orangish-Brown because cognac is a deep golden-orange color. "
Finally, a description of the intensity of the colour is added, including " Faint, Very Light, Light, Fancy Light, Fancy, Fancy Dark, Fancy Intense, Fancy Deep, Fancy Vivid."
A certified gemologist can give you the description of the gem in question for a fee, which will include its full descriptor name.