Prior to the fixed term parliaments act, Prime Ministers were (effectively) free to choose the timing of the election. The Parliament act required that an election would be held at least every 5 years, but holding an election between 4 and 5 years would not be considered a snap election. 

There were snap elections in 1923, 1931, 1951, 1955, 1966, two in 1974, but none since then.

There are various reasons for the PM to call a snap election: Generally, you can divide the elections into those where the PM had no choice but to call an election, and those where the PM was seeking some political advantage.

In several of these cases (1951, 1966 and Oct 1974) the parliament was either hung, or the Prime minister had a majority of less than 10 (which is not considered to be a "working majority" in the UK's parliamentary system) and so you can say that they "jumped" to call an election before they were "pushed" by losing a vote of no-confidence. In 1931 there was a governmental split caused by the great depression, in which the Labour prime minister lost the support of his own party.

The PM was seeking political advantage in 1955 after Churchill retired, and his replacement was seeking a personal mandate.  And in 1923 and Feb 1974, the Prime Minister was seeking a renewed mandate to enact specific policies (tariff raising in 1923 and to deal with a miners' strike in 1974)

So "snap" elections are not historically rare, but there haven't been any for over 40 years. For an incoming president to seek a special election would be impossible, since it is not provided for in the constitution. The UK doesn't have a written constitution. And snap elections have clear precedents.