Learn Swedish: Kan du köra den här trucken hit? - Can you drive this forklift here?

Because there is movement involved! Hit is a directional adverb that means 'to here'. Since you are asking someone to drive the forklift to your location, you must use hit. You would use the positional adverb här if the forklift was already parked and stationary.

But the sentence also uses här earlier on. Why doesn't the movement rule apply there?

In the phrase den här, the word här is not acting as an adverb of place. Den här is a fixed phrase that simply means 'this'. It just points out which forklift you are talking about, so it isn't affected by whether the vehicle is moving or not.

Why is it trucken (the forklift) and not just truck when the English translation says 'this forklift'?

In Swedish, when you use the pointing words den här (this) or den där (that), the noun following them must always be in its definite form. You are literally saying 'this here the-forklift'.

Can I use the word truck to mean a pickup truck or a large delivery truck?

No, truck is a bit of a false friend! In Swedish, a truck almost always refers to a forklift or a similar small industrial vehicle. For a large truck driving on the road (a lorry), you would use the word lastbil.