If you check out the Wikipedia page on Properties of Water, it has a very small history section which says Cavendish (not a chemist, arguably) determined the composition of water in 1781. Then following that lead to Cavendish's article, you'll learn quite a lot, unless you already knew it. As far as us putting "the complete picture" in front of you on a plate, yeah, that's not going to happen. We'd have to know what your background was, and even then what exactly would satisfy your concept of what the "complete" picture was. The up-side is that although satisfying yourself is up to you, the web has a lot of resources you can get help from. When it comes to a lot of scientific, mathematical, and technical subjects, though, you'll have to put in the study (and practice) time to learn and understand it, course work and textbooks will be difficult to replace with unorganized web research. OK, enough philosophy. I encourage you to put together a time-line of alchemy, natural philosophy, and science at around the start of the 19th Century. You probably know that the idea that matter is composed of small indivisible particles (atoms) was first seriously considered by the ancient Greeks, right? And that various technologists have used recipes (based on weights or volumes or both) to repeatedly create (or destroy) various materials of interest, from time immemorial. Alchemists were dedicated transmutation, so what they knew and when they knew it is important here. Proust didn't demonstrate his law of definite proportions until almost 20 years after Cavendish discovered the constituents. So first we knew what it was made of (oxygen and hydrogen, although the concept of elements was not yet firmly established.) And see also https://en.wikipedia.org/wiki/Oxygen#Early_experiments as well as the following discussion on the Phlogiston theory. Probably the reason you haven't gotten a clear answer here is because this isn't a technical question, it is a historical question. To get a clear complete "picture" you need to literally waste your time learning about all sorts of discredited theories (Phlogiston, and the earlier even more unfortunate Earth, Air, Fire and water "theory") and then learn how the various people overcame the blunders made before them to find the "right" explanations for the world around them. Needless to say, that is an ongoing effort but to recapitulate all of the dead-ends isn't something we're going to do in a short answer, frankly. Anyway, put together a time-line and you'll begin to see how the various strands of the story wove together. HTH.
