To be precise, on international flights (except North America) BusinessFirst will continue.
Originally Posted by United website
United Global First - United Global First will be the name of our first-class product on three-cabin aircraft. You’ll get a higher level of privacy and the luxury of a flat-bed suite, personalized attention and priority airport services.
United BusinessFirst – Our business-class product is coming to the United fleet, which will elevate business-class services and comfort for long-haul international flights. United BusinessFirst will be the name of the front cabin on two-cabin aircraft and the middle cabin on three-cabin aircraft.
United First - United First will be the front cabin for customers traveling within the U.S., including Hawaii and Alaska.
United Business - United Business will be the front cabin on all other routes. This includes flights operating to and from Canada, Mexico, the Caribbean, Central America and northern South America, as well as intra-Pacific routes where United Global First and United BusinessFirst are not available.
Fare buckets as you say are a completely different matter.