Is "medium white sauce" a usefully distinct term?
After spending more time on search engines and thanks to the comment from belisarius...

Medium refers to the thickness level of the final sauce, and is controlled by the ratio of roux (butter/flour) to milk. For 1 cup of milk:


thin = 1 tablespoon each flour/butter
medium = 2 tablespoons each flour/butter (a "standard" bechamel)
thick = 3 tablespoons each flour/butter


Sources: 1, 2