There are perfectly clear definitions of what a firearm is.
However, when politicians, keen to be seen to be doing something, give the results of a "firearms" amnesty, they are keen to include anything handed in under the definition of "firearm". The press are also keen to portray anything remotely resembling a gun as a "firearm".
Strangely enough, so are gun lobby groups in America, when it comes to the British statistics.
As to crime with firearms, a firearm used in a crime is anything the criminal presents as a firearm. So if you walk in to a shop with two pieces of pipe sticking out of a bag, (which would look like a sawn off shotgun) and claim it is a shotgun, that's an "armed robbery".