There was a big debate about this in my coin club (which is called the "Queensland Numismatic Society", so the matter of definition is kind of relevant); it was before I joined, and sparked by certain members taking up meeting time with discussion about things that some other members deemed were "not numismatic". The definition which resulted from the debate was as follows:
Numismatics is the study of coins, and objects that are related to coins either by physical appearance (eg. medals), monetary function (eg. tokens and banknotes) or method of manufacture (eg. badges and seals).
It should be noted that "numismatics" and "coin collecting" are two separate, although usually overlapping, things. The primary goal of a coin collector is to collect coins; acquisition and accumulation are the main motivators. The primary goal of the numismatist, on the other hand, is discovering and recording knowledge about the coins. Many people are, of course, both. But it's entirely possible to be a "numismatist" and not actually own a single coin.
