So it's like saying that the expression:
"Bob says we cannot play baseball without a bat or a ball, since he says that we need a bat and a ball to baseball." really means that we can play baseball with either a bat or a ball.
No, I disagree completely. It is not like that expression. The keyword was underlined
or. In your translation you did not change this word. The meaning differs little in your translation from the other. In the expression, the word "or" is what implies that you may play baseball with "either" a bat "or" a ball. Clearly, in the translations the "or" was not the relevant issue.