You seem to have a reasonably OK understanding of what he "did." What you are actually trying to get at is an interpretation of the importance of what he did.

How do we measure importance? With Turing there are two common metrics:

• The importance to World War II

• The importance to the history of computers

• (We could also add "the importance to the history of mathematics" but that is getting into rather arcane territory.)

For the former, you dismiss his work as "merely a codebreaker." I am not sure what you mean by that, but presumably you mean that he was just part of a larger team of people whose jobs it was to break Axis encryption. This is true, but not exceptional — all "great men" work in contexts of other people, though sometimes we tend to ignore them. Turing's work here was to help generalize the Bomba approach to cracking the Enigma code. He didn't necessarily implement these generalizations himself, nor was he the only one to contribute to them, but in my understanding he was pretty fundamental in making this happen from the design/theoretical point of view. This contributed towards the cracking of the Enigma and towards the winning of the Battle of the Atlantic.

As for his importance to the history of computers, his biggest contribution was the formalization of the idea of a "universal computer," showing that custom-built logical processes (that is, computers that were built to only solve a specific problem or class of problems) were inefficient in that a properly developed logic system could be used to solve all problems (limited only by time and resources). This is a fairly interesting development along the way to the universal computers that we use today, and a marked difference from the kinds of computers that were being developed during World War II, which were "specialists" in their computational capabilities.

Now, if you are asking, at the end, whether we give Turing credit for all of this, or whether we credit it to World War II as the context — this is a bigger historical question that you can ask about anybody in history. The only sane answer to me seems to be that these things are interrelated. Context is an unavoidable and necessary precondition for anything. Could Hitler have been important without World War I? Could FDR have done what he did if not for the Great Depression? Surely there can be no Churchill without the British Empire? Etc. The existence of context as a precondition does not lessen the contributions of the individual within that context.

I find if you want to gauge relative "importance" a useful thought experiment is to ask, "if this person had been removed from history, what might be different?" Obviously it is impossible to know this with any certainty, and it gets into dangerous historical ground. But this kind of counter-factual can serve to focus the mind around the places where the person contributed. If we remove Alan Turing from the equation, do the British get invaded by the Germans? Probably not, but perhaps things are a little bit harder. Does the development of the computer slow down any? Probably not — but this is in part because Turing's actual contributions were more theoretical than practical, and many of them were kept secret until the 1970s, so they did not contribute strongly to the key developments that took place in the 1940s-1960s.

You might separately want to ask yourself, why do people talk about Turing? Part of this is because we are clearly attracted to charismatic individuals with interesting stories, and Turing surely was that. The tragedy of his story is part of our interest in him, as well — Turing has also become a moral lesson (e.g. "prejudice will destroy people of great value").

As per the assessment of the importance of his work, not to repeat the points well-made by restricteddata, i want to add the view on Turing from the inside of computer academia.
Turing postulated the problem of "Computability", that is, the realm of problems that are solvable by a computer, and demonstrated that any mathematical problem is solvable by a computer.
In computer science, a thought experiment known as "Turing Machine" is often made, where a machine with a limited instruction set, that can manipulate a one-dimensional, infinite-length tape, has to solve a particular problem.
It's been proven that, if said Turing Machine can solve a problem, so can a modern computer - and the other way round: If the Turing Machine can't, then neither can the modern computer.

What does that mean, for the man's legacy?
He, for the first time, postulated exactly what kind of problems could be solved with a computer and which ones couldn't.

The Turing machine in my view was his great achievement (I have a love of maths). I have had discussions before where people see the "machine" as that rather than a thought process; which is missing the point. The machine was never meant to be created literally, rather a thought device to prove or disprove certain constructs. He used it effectively to disprove the "Enstscheidungsproblem." That particular problem was the last of the three challenges posed in 1928 by David Hilbert to the world of mathematics. (The other 2 were solved by Godel - who is also deserves massive amounts of respect - Hawking called him the greatest mathematical logician of all time). Turing's approach to the solution was revolutionary. It is reported that he was actually thinking more in terms of the typewriter rather than a computer when he proposed the Turing machine.

I often think about this thought device when evaluating problems to do with large amounts of data. (e.g am I creating a self referencing logic loop).

(Source: God Created the Integers - Stephen Hawking)

The other aspect is that as far as I am aware we don't know if all of his work has been released from secrecy acts. When I studied complexity and computability there was rumours that his best work (crypto) will remain withheld for a very long time. No sources, just rumours in maths circles. There was additional releases in the last decade (sorry 2012) or so. If he was merely a codebreaker, the British government wouldn't have held onto his work for so long (70+ years).

*edit to add the 2012 part / changed prove to disprove. Its late; sorry.

Theoretically, Turing developed one of the three (equivalent) descriptions of 'effectively calculable'---the principle basis for computing. (The other two definitions came from Church and Goedel.) Turing's formulation of effectively calculable was especially important as it provided a modern description of not only computation but also computing machines. Goedel and Church's descriptions were either not constructive (Goedel) or not immediately amenable to implementation (Church).

See Wadler's 'Propositions as Types' for an overview of the history of effectively calculable equivalence.

Which is lesser known, Alan Turing also did some work in biology, especially morphgenesis. Here is the scan of his paper The chemical basis of morphogenesis