I thought possibly you would enjoy what the British, themselves, write on the subject. This comes from the print version of the Encyclopaedia Britannica, which is, for all purposes, the final word.
"The names United Kingdom, Great Britain, and England are often confused, even by U.K. inhabitants. England is just one country within the Kingdom ... while the United Kingdom also includes Northern Ireland. Wales and England were unified politically, administratively, and legally by the acts of union of 1536 and 1542. In 1707 Scotland joined England and Wales in forming a single Parliament for Great Britain, although the three countries had previously shared a monarch."
"Great Britain, the island comprising England, Scotland, and Wales, forms together with numerous smaller islands, the archipelago that is as irregular in shape as it is diverse in its natural heritage."
"The traditional division of Great Britain is into a Highland zone and a Lowland zone."
"The main watershed in Great Britain runs from north to south, keeping well to the west until the basin of the River Severn."
"Great Britain's marginal position between the European landmass to the east and the ever-present, relatively warm Atlantic waters to the west, ensures the modification of both the thermal and moisture characteristics of the principle types of air reaching the country's shores."
You see, the references to "Great Britain" are in the geopgraphical sense. If you were to read the article, when reference is made to the government or the country, the term "Uninted Kingdom" is used. If you don't consider the Britannica, to be authoritative, then possible the Oxford History of England series would help:
"The name 'Great Britain' [as compared to the former 'Britain'], was a reminder of King James I, as was the holding of Scottish peerages by Englishmen like the Fairfaxes and Falklands, and of English peerages by Scotsmen like James Hay, earl of Carlisle."
Another encyclopedic entry explains: "The term Great Britain was first widely used during the reign of King James VI of Scotland, I of England to describe the island, on which co-existed two separate kingdoms ruled over by the same monarch."
It should be clear to you now, that the reason why the first and preferred usage of the term "Great Britain" refers to the island upon which the United Kingdom is found.
Don't feel too bad, though, because as the Britannica notes, even the British screw it up too.
According to who? The Grand Chief Poobah and Arbiter of Dictionary and Encyclopedia definitions?
As I duly noted and documented for you, the term "Great Britain" derives from the official name of the country we know as the UK: "The United Kingdom of Great Britain and Northern Ireland," as called by the UK itself and by the US government in recognizing the UK.
The proper historical term for the island on which Loegria, Cambria, and Albany are located is Britannia as it has been referred to since at least ca. 1136 when Geoffrey of Monmouth standardized the term, along with Britain's supposed geographical history, in his Historia Regum Britanniae. Thus we get Britanniae, which translates into the designation Britannia.
And yes, considering Geoffrey of Monmouth's virtually unparalleled historical contribution to the development of the British culture and identity, i'll take his word on it over both the Encyclopaedia Britannica and the Oxford reader on any given day of the week.