Raum is a homonym. It's either space or a room. There is no difference between Raum (room) and Zimmer in daily usage, although there is a formal difference: a Zimmer is a room designed for housing people (bedrooms, living room, etc.), while Raum (room) can describe other rooms, too. Bathroom, Kitchen, etc. Stube is a rural, but mostly archaic word for living room. Der Raum which translates to space is a whole different thing and part of some other words like Weltraum and Freiraum.
Raum is not really a homonym, it just has the sense of space in general as well as a room (a defined space or chamber, usually within a building).
Homonyms are two words with the same spelling and pronunciation, but different meanings origins. E.g., stalk is the main stem of a plant, but also the act of following or harassing.
Correction: Homonyms either have the same spelling or the same pronunciation. Homonym is a broader term for either a homophone (sounds the same) or homograph (spelled the same). Etymologically, these words come from Greek bases. HOMO = same ONYM = name/word PHON = sound GRAPH = writing. So, both homophones and homographs are homonyms. Just wanted to clarify to avoid confusion.
Given the vast number of people who contribute and care deeply about their own field of expertise, I would expect that one of them would have by now corrected the Wikipedia article on homonyms which leads off with:
In linguistics, a homonym is a word that has different meanings. In the strict sense, one of a group of words that share the same spelling and pronunciation but have different meanings. Thus homonyms are simultaneously homographs (words that share the same spelling, regardless of their pronunciation) and homophones (words that share the same pronunciation, regardless of their spelling).
One would also expect the Wiktionary article (http://en.wiktionary.org/wiki/homonym#Noun) would also have been changed by now.
This illustration might help to illustrate:
So, some (but not all) homophones are homonyms, and some (but not all) homographs are homonyms. There is not, however, complete overlap.
You're right. But, according to the Merriam-Webster dictionary, a homonym can also be:
one of two or more words spelled and pronounced alike but different in meaning (such as the noun quail and the verb quail)
Raum can also be used in the astronomical sense.