However, the name "Britain" (not Great Britain) includes Northern Ireland, for which reason "Britain" has the same meaning as "United Kingdom".
But to understand why "United Kingdom" is in its constitutional understanding a Union of three Kingdoms, see also the Union Jack (synonym for Union Flag), which has the national Emblems - in the sense of Crosses - of England, Scotland and Ireland. But not of Wales (which is a dragon). The Union Jack also has the Blue of the Scottish Flag as background.
Now really, did Nigel Farage know this, before tooting out in Scotland?
That he would have to change the composition of the Union Flag if Scotland left the Union.
But then UKIP uses the £ symbol.

(we pay 50 of 'em ev'ry day to the EU).
And see also the term: Prince of Wales, historically referred to the heir to the English throne.
The English Monarchy under Henry the somethingth (3rd or 4th maybe) introduced this title to appease the Welsh, whose Country had just been incorporated into the English Kingdom.
Apparently it gave the Welsh a feeling of .... something ... importance, perhaps.
The reason the English Monarchy did not consider Wales a kingdom, was because they found the Country, on entering it armed with mace and shield, as a federation of principalities, independent from each other.
The Welsh dispute this was the case, though.
Actually it was a Norman aristocracy that conquered Wales, probably the same which had conquered England (not Kent) some generations earlier. Kent received the title of "Invicta" for not surrendering to William the B..., but rather coming to an agreement with him.
The English also nicked the Welsh longbow and used it against the French and the Scots with devastating effect.
However, the Scots retorted with the pike, to stop the English cavalry charges.
Finally, the Saint Patrick Cross is currently not the national Emblem either of the Republic of Ireland, or of Northern Ireland. Northern Ireland uses the Saint George Cross with a hand in the middle.
So, in conclusion, without Scotland, there would be one Kingdom (England), a Principality (Wales), and part of a Kingdom (Northern Ireland). A principality is not a Kingdom, so we would lose our title of "United Kingdom", our Union Flag, our incomes (Scotland 's oil and gas and Scotland's corporation taxes), and be lumbered with Nigel Farage.
Now back to my milk.