I was watching a Japanese video and it seems it is more accurate to say 傘を持った rather then 傘を持っている to indicate that I am holding an umbrella. For example if someone asks "傘を持った？" the grammatically correct reply would be "持ったよ" rather then "持っているよ". Why is this so? To me, 持った sounds like I held the umbrella (and don't have it now), whilst 持っている sounds like I am holding an umbrella now. Therefore, 持っている sounds more correct, but that doesn't seem to be the case.
傘を持った。 can mean having it in your hand or having a folding umbrella in your bag.
Since you are asked 持った？ we answer back 持ったよ。
If you are asked 傘を持ってる？ we answer back 持ってるよ。
傘を持っている？ is often used when it is raining outside and you are not sure the person is having an umbrella in their bag.
傘をさす。 To put up an umbrella