Cup and cone is proper bearing terminology. "Race" is something that a non-bearing person would call it. (Technically a race is the surface that the bearing rolling elements run on, not the whole part).
Unless you have new rotors (with new cups), you need to replace the cups and cones both. Do not mix old and new parts as the parts wear together.
I recommend sticking with Timken, ***, or SKF bearings which are made in US, Germany and Sweden. There are lots of cheap bearings out there that are made in China or Poland. NTN and Koyo also make good bearings if you want a Japanese product. Bower is a crap shoot as far as country and quality.
Technical point - tapered roller bearings are available as seperate cups and cones because each bearing is available in multiple ID's and OD's and you simply find matching cups and cones with the dimensions you need with the load characteristics you need (different taper angles and series give different load characteristics). Metric tapered roller bearings however only come in sets (There are probably less then 20 metric sets compared to thousands of inch dimension bearings).