I am immediately reminded of a great 2010 documentary called Top Secret Rosies: The Female 'Computers' of WWII.
The word "computer" has had a definition since at least the 17th century. In the traditional sense, this from the Oxford Dictionary of English Etymology (1992), "compute" came from French, meaning "determine by mathematical reckoning." The word computer would refer to a profession wherein one makes their living doing mathematical and arithmetic calculations. The Wiki on human computer should give you loads of insight into this old profession. An nice additional piece of historical perspective is encapsulated in the late 19th century job advertisement, A Computer Wanted.
Historically, when some large computations for some large complicated process needed to be done, and the person wanting them done had the cash to hire the people, human computers would be hired, organized and instructed to take on various parts of long computations in an organized assembly line style. Computer was still a human profession right in to world war II, and the first electronic computers like ENIAC were actually programmed by these human computers. The people who were in this profession typically had math degrees.
ENIAC was not the only electronic computer of the 1940s. An earlier computer, Colossus, was used by British cryptologists during the war, and the Bombe computer was used to help decipher the German Enigma machine.
I also recall Richard Feynman telling stories about his days working on the Manhattan Project. In Feynman - Manhattan Project the author states, "The greater part of his work was administering the computation group of human computers in the theoretical division." Much more first person anecdotal stories of his time here working with his human computers, as I recall, can be found in his book Surely You're Joking, Mr. Feynman! That is a much recommended read, I do not possess a copy; I read it in a library. I vaguely recall him describing his process as one where these students in an orderly fashion, an adder, a multiplier, a divider, and so forth, would, with the use of a punch card system, build up big computations over many hours and days, maybe even weeks. One human misstep would obviously have disastrous results. So I guess he was sort of using a combination of human computers, and machine style punchcard devices.
Essentially, the answer to your question is (mostly I think) the latter; electronic computers arose straight from a desperate demand for more computational power. Being able to automate a huge algorithm formerly done by a team of directed humans in less time with less error became possible, and it was important at the time. They were really pushed along out of necessity in a big way by the demands of the second world war, and the cold war funding immediately thereafter kept this advancement moving forward. I am not saying that WWII gave us the very first examples of computers, but this would be the age when they were dramatically ushered in, and the word computer really began to transfer its meaning from "one (a person) who computes," to these machines that, in a decade, made the job of human computer obsolete, and eventually turned into that thing currently sitting on your desktop.