Correct it originally meant someone from Great Britain, the largest of the British Isles, not England specifically (as you noted, but touched below); it’s specifically referring to the land mass consisting of England, Scotland, and Wales. It technically doesn’t include North Ireland, and thus was not referring to a person from the UK, but it colloquially did, and thus became over decades of the union to have the same meaning, though subtly different grammatical usage, as British person (below).
Referring to Britons as the Celtic Britons (which were not just England) is only used in a historical context, otherwise it’s considered an archaic usage of the word.
British originally meant “from the British Isles”, which includes all of Ireland. So by that definition, someone who is Irish, is British … and now you’ve just been punched in the face.
British as since moved its definition away from the British Isles itself and just refers to The UK, The British Overseas Territories, and The Crown Dependencies. If you otherwise qualify for a British Passport (ignoring the legal and criminal stuff), you are British, or you are a British person, or you are a Briton.
By modern usage, and preferred grammar.
You’re right that Briton isn’t commonly used, usually due to a mix of it being a homophone, and it not really being taught. The colloquialism Brit, is by far the most common; however, it is a colloquialism. Briton it is the grammatically correct, noncolloquial word.
On a side note, rather than using an American dictionary for an English thing, it would be better to use the English Oxford or Cambridge dictionaries. Not only is American English the odd dialect in all the major dialects of English for grammar and spelling; I have found in the past dependencies between a word or phrase specifically English, and specifically American English. Thus when I know something is specifically one or the other, I’d refer to that specific recognised dictionaries, and falling back on an English dictionary if it is neither – considering I am English for no other reason.