持つ and 運ぶ: hold something or transport it
Distinguish holding or possessing with 持つ from moving an item to another place with 運ぶ. Read the direction of 持っていく and 持ってくる from the conversational reference point.
持つ: hold, carry, or possess
持つ can describe holding an item in the hand or keeping it with you. 右手に傘を持っています。 means that the person is holding an umbrella in the right hand, and 傘 takes を as the item held.
持っています can also describe possession. 車を持っています。 normally means owning a car, not lifting it by hand. Use the kind of object and any location phrase to determine the intended sense.
運ぶ: move something to a destination
運ぶ focuses on moving a person or object from one place to another. この荷物を二階まで運びます。 means taking the luggage to the second floor. The sentence alone does not say whether it is moved by hand, cart, or vehicle.
箱を持っています。 can describe simply holding a box, while 箱を倉庫へ運んでいます。 describes transporting it to a warehouse. Do not treat temporary possession and the task of transport as the same event.
The reference point in 持っていく and 持ってくる
持っていく combines the te-form 持って with 行く and presents an item moving away from the current reference point toward another place. 会議室に資料を持っていきます。 means taking the materials to the meeting room.
持ってくる presents an item moving toward the speaker’s location or another reference point. この書類をここに持ってきてください。 asks someone to bring the document here. By contrast, 机を隣の部屋へ運びます。 focuses on the task of transporting a desk to the next room. Check both direction and whether the sentence focuses on bringing an item or on transporting it.
Quick check: Which fits a request to bring this umbrella here, 持っていって or 持ってきて?
Show the answer and why
Use 持ってきて, as in この傘をここに持ってきてください。 The umbrella moves toward ここ, the conversational reference point.