That's because the plant creator registered it as a series and a trade name, the trade name is 'Red' in that series. The second one you posted is with just the cultivar name, it was not created with a separate trade name.
When you enter a new plant, it's explained: "Trade names are generally trademarked names that sellers use for marketing the plants. A plant with a trade name will always have a cultivar name (often a series of numbers and letter that don't spell anything). The trade name almost always includes a (tm) symbol. When you have a group of different but similar plants that have trade names, then you can group them by using Series."
The begonias you listed don't have generic entries per se, they are species names followed by cultivars related to those species. So we have two wax species listed so not all wax begonias belong together in one entry, depends on which species they are related to. Same thing with the cane ones.
For wax:
Wax Begonia (Begonia cucullata)
Wax Begonia (Begonia x semperflorens-cultorum)
Cane:
Cane Begonia (Begonia coccinea)
Cane Begonia (Begonia aconitifolia)
My bad! The Reiger's do have one generic entry:
Rieger Begonia (Begonia x hiemalis)
As does Rex:
Rex Begonia (Begonia rex)
