From a data compression/language efficiency standpoint, both sentences in both languages actually rely on a (potentially large) amount of unstated context to sort out these ambiguities. In some languages, this context can be totally unspoken and merely known to both the speaker and the listener. This absolutely MUST be accounted for if a truly correct translation is to be made.

For instance, your assumption that the definite "the cat" is being used idiomatically like so: this sentence, used in the manner you offer, might be used in conversation might occur in a farmhouse somewhere between an old man and woman who have lived together in this house for a long time, i.e. American Gothic. There's a vast amount of shared information and a perception of very little ambiguity held by both the speaker and the listener (whether correct or mistaken!). Any of those might fail. Furthermore, to use this sentence in English unadorned by context requires that both the speaker and listener have a shared reference to _what_ cat is being referred to by the definite article, "the". This very well might come with an unambiguous default in other languages!

Translation only gets more complicated from this.