Yeah, but I think we borrowed it from Italy ("parasole"), where you probably more often need an umbrella to hide from the sun than from the rain. And actually, the English word also has nothing to do with rain and stems from the Italian "ombrello" (Italian "ombra" = Latin "umbra" = "shade", see).
Added, but I don't see the interchangeability of those words in Polish... Sometimes maybe the difference between is not important, but they still mean different things.
"Zawsze bierze jakieś ciasto na imprezę" vs "Zawsze przynosi jakieś ciasto na imprezę" (she takes, vs she brings. the result is the same, but I wouldn't call them the same).
I disagree that it's clearly the mother's umbrella. Maybe she's stealing it, or borrowing it, or confiscating it or any number of situations where someone taking an umbrella doesn't imply possession. It makes sense that it was marked wrong because the sentence you're asked to (directly) translate has no indication of possession
Simply because it's a correct answer. Let's change "umbrella" to a "wallet" - usually every person has their own wallet, that's more clear than with an umbrella. "My mom is taking her wallet" can easily translate to "Moja mama bierze portfel" because it's quite obvious that she's taking her own wallet.
Here it is "just accepted", but with family members, you can observe the same difference in the main answers. "I am talking to my sister" is most naturally translated as "Rozmawiam z siostrą", without a possessive, because I usually talk with my sister, not with someone else's sister. If it was someone else's sister, then I would specify it, but I won't mention the obvious thing that this is my own sister.