"We are cutting the bread with a knife."
Translation:Kroimy chleb nożem.
So "z" is mostly used to convey "with (someone)" or when you are pairing two separate things, as in "X with Y."
It is not used to convey (one of the meanings of) "with" that is used in English, which is "by means of." In Polish, the sense of "by means of" is conveyed by following the verb with the "means" stated in the bare instrumental case, without any intervening preposition.
Am I understanding this correctly?
It's never optional, it just changes the meaning and using it (or lack of it) wrong results in absurd sentences.
"z" is simple "with", mostly "with someone", but also "ryż z jabłkami" (rice with apples) and similar.
Instrumental without "z" means that something was used as an instrument. "kroić nożem" = to cut with (using) a knife. "kroić z nożem" = to cut, while the knife is also cutting (as if it was animated).
Similarly: "Jem zupę łyżką" = I am eating soup with a spoon (using a spoon), vs "Jem zupę z łyżką" = I am eating soup and the spoon is sitting next to me, also eating soup.