Artful Codger - It just happens to be the linguistic fact that, in the case of murderers, the weapon is idiomatically linked to the deed, as in 'axe-murderer'; whereas, in the case of assault or corporal punishment, this is not the case - nobody would have called an old-fashioned schoolmaster a 'cane-beater', but a 'caner', or a 'caning teacher'; because it was not the cane that he beat, but his ill-behaved pupils, using the cane as his instrument.

Maybe the language is inconsistent in this; but I insist that such is the idiomatic or colloquial fact of the language nonetheless.