I disagree with the claim that "Dragon Wing" is a common name for many begonias.
It is the term of "Angel Wing" that is commonly used for many cane begonia species and hybrids with wing-shaped leaves (including Dragon Wing® Begonias).
The Dragon Wing® name is registered to Ball Seed/PanAmerican Seed for their two interspecific F1 hybrid seed strain cultivars of Begonia rubriflora (U014) x Begonia semperflorens, called 'BEPAPINK' and 'BEPARED'. Only the F1 seed from this cross is viable. The seedlings/plants that grow from them will be sterile and can't set seed. That's what makes them such non-stop, blooming machines. Great plants!
(Although, they can make a right slimy, slippery mess on your porch if you don't keep up with sweeping away all of the fallen flowers!)
Dragon Wing Begonia (Begonia Dragon Wing® Red)
Dragon Wing Begonia (Begonia Dragon Wing® Pink)
Interestingly, although
Begonia 'Canary Wing' was developed from a branch mutation of 'BEPAPINK' it can't be marketed as a Dragon Wing® Begonia. Since the mutation wasn't found/developed by Ball Seed, they don't own the patent so they have no reason to share their trademark.
The Dragon Wing® Begonias have been so popular since they were introduced around 20 years ago, that it was probably inevitable that the "Dragon Wing" name would get misapplied to other, similar-looking Begonias on occasion.
There are also a couple of earlier hybrid cultivars called Begonia 'Dragon Wing' and Begonia 'Dragon Wings' but these are unrelated to 'BEPAPINK' and 'BEPARED'. I'm not even sure if they are commercially available anymore.
Like Tiffany said, cutting your Dragon Wing® Begonias back partway will do no harm and will also give you cuttings to root!
Begonias are the best!
