I did not look up the definitions, but do think I have a (probably rather subjective :-)) concise interpretation of the definitions you asked about. I mostly do agree with 0x6d64, but not completely. As far as usage of words is concerned (referring to the definitions you asked about explicitly).
Assuming you are not a native German speaker, then if a word is marked in Duden as:
- umgangssprachlich: you are safe to use these words in almost any context (possible exceptions: you give a talk at a formal event and you are supposed to use formal speech, address a king,...) Everyone will understand you and usually no one will feel offended.
- derb: Simply don't use them, unless you are absolutely sure you know everyone very well and misunderstandings won't cause trouble. It can almost certainly be used to be offending by intention if (mis)used appropriately. In certain communities words marked as 'derb' in Duden, for example, may be very common, though. From their point of view they may rank as 'salopp'.
- familiär: Difficult to explain, as a matter of fact. I'd not say it's use should be restricted to addressing kids or family members, but a German would use it only when addressing people he feels, somehow familiar with. This may include neighbours, relatives or coworkers. Often these words are used to talk to people close to you in a nonoffensive way about things for which you only find rather technical -if not scientific- terms or very offensive ones, e.g. when talking about sexuality. Words marked as familiar may possibly be used only by certain communities or in certain regions, and it may happen that people don't understand you if you use them in a community which does not. Usually this won't be regarded as offensive, though. Words marked as familiar won't be commonly found in the news, newspapers, scientific or other publications. You'll hear them in the street or on TV in your daily soap, though.
- salopp: not adhering to conventions, without being offending by intention. This means that some very conservative people might feel offended, while most won't. If a word is marked as "salopp" I'd advise to avoid using it unless you know who is listening and how they will feel about it. If you are among native Germans and they use such words regularly, you are probably safe to use it in their presence, too.
Unfortunately it's very difficult to classify words using these definitions. Words marked as 'derb' may be regarded as 'salopp' or 'familiar' in certain regions of Germany, while they might well be considered as rather rude in other regions. This may be the reason for classifications which are hard to explain.