Briton is the most widely accepted term for people from Britain (which of course is not the same as England and the United Kingdom). Britisher had a brief heyday in the 20th century, but it was always only an American term and was never accepted by Britons themselves. Brit is not offensive, but it is informal.
Of these words, Brit appears most often because it serves as both a noun and an adjective, and probably also because it’s short and not a homophone with Britain. Briton is only a noun, and it only denotes people. Britain is always the correct spelling of the place name.
Examples
Here are a few examples from both British and non-British publications:
Sir Howard Stringer is the Briton who chose to serve in Vietnam so he could stay in the United States. [Guardian]
An expatriate Briton, Hadley was a Roman Catholic convert who devoted his life to society’s casualties. [Montreal Gazette]
In 2000, he became the first Briton to win the Hans Christian Andersen Illustrator’s Award for services to children’s literature. [Financial Times]
The 68-year-old Briton used to get so nervous before shoots that he’d vomit. [Forbes]
He is also understood to be the youngest Briton to stand atop the Himalayan summit. [Evening Standard]
“No he wasn’t,” the Briton scoffed. [Stuff.co.nz]