In the instance of the war on Iraq, there certainly were many reasons why many people wanted war. The WMD one was both one of the scariest for the general public and the least likely in practice, though in some fairness since the USA had sold those weapons to Iraq it had reason to believe that Iraq still had them.
So, yes, in general terms, the reasons for war can be quite different to the stated reasons.
The Versailles Treaty that effectively ended the First World War put Germany in an almost impossible position, with the Great Depression making it impossible. Few people in the world, from General Smuts in South Africa (possibly the most outspoken) to the Germans themselves felt that it was sensible, or even fair. Had Germany realised the terms of peace, it is unlikely they would ever have called a ceasefire but by then it was too late: they had effectively capitulated and their enemies were even more prepared for war. Had it been a small country like Belgium demanding such punitive damages, other nations would have been able to talk them down. As it is, France demanded harsh penalties and their allies had little option but to go along with it or continue the war, which they didn’t want, either.
So when Germany started flexing its muscles and breaking the Treaty, many thought it understandable. While the annexation of lands from other countries was not welcome, there was general understanding that most people in those lands wanted to be part of Germany and in the spirit of the League of Nations, where people are supposed to be able to decide such things for themselves (one of the aspects that was modified when carried into the United Nations) there was little to complain about.
The British had received assurances from Germany that Germany had no interest in land other than that which was clearly Germanic. While that might be naive with the hindsight of history, there were reasons to believe in it (nobody took Mein Kampf very seriously: a boring book written by a corporal in prison was common pulp at the time and a politician was not expected to do what he said he would do).
So when rumours of the intent to invade Poland came through, the British were jolted from an uneasy acceptance of Germany’s right to take German lands, and the hope that it was all that Germany ever wanted. Suddenly, very non-German territory was under threat and this time, it was to a country considered a friend, and a country that had seen a great deal of the fighting only a few decades earlier. Furthermore, any encroaches by Germany into Poland were likely to be matched with Poland being split once again between Germany, Russia and Austria (Austria/Hungary) potentially starting ‘The Great War’ (World War 1) all over again.
Britain’s ultimate reason to go to war on Nazi Germany might be stated as being the need to curb German aggression against non-belligerent nations. To that extent, it didn’t matter if Germany went for Poland, Denmark or the Netherlands (all of which they could make some kind of claim for), the British response would have been the same. If Germany was going to try taking over the world, war was inevitable and Britain simply knew that sooner was better than later.
As it is, Germany turned its eyes first on Poland and Britain & France showed the ultimate support for that nation by demanding that Germany get out or be at war. Of course, physically Poland is remote from Britain & France and to that extent only it was ‘alone’ because none of its bordering nations gave it the backing it needed. However, Poland the nation, created after the First World War and supported economically by the British Commonwealth, did not stand alone when attacked. Poland was supported as much as was possible by Britain, the British Empire, some British Commonwealth nations, France and the French empire.