fahrend
moving
Describes a vehicle or object that is currently in motion.
Das Aufspringen auf einen fahrenden Zug ist extrem gefährlich.
Jumping onto a moving train is extremely dangerous.

moving / traveling
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moving
Describes a vehicle or object that is currently in motion.
Das Aufspringen auf einen fahrenden Zug ist extrem gefährlich.
Jumping onto a moving train is extremely dangerous.

'Fahrend' (moving/traveling) is the present participle of the verb 'fahren' (to drive/travel). Such participles describe an action currently taking place and are often used as adjectives.
Ein schlafendes Kind soll man nicht wecken.
You should not wake a sleeping child.
This word is used almost exclusively as an adjective before a noun, e.g. 'der fahrende Bus' (the moving bus). Using it as a standalone predicate adjective, e.g. 'Der Bus ist fahrend.' (The bus is moving.), is very uncommon and sounds unnatural.
Er winkte dem fahrenden Auto hinterher.
He waved after the moving car.
You can recognize the meaning from the noun: if 'fahrend' (moving/traveling) refers to a vehicle or object, it means 'in motion'. If it refers to people or groups, it means 'without a fixed residence' or 'traveling from place to place'.
Ein fahrender Händler verkaufte Schmuck auf dem Markt.
A traveling merchant sold jewelry at the market.
In the meaning of 'traveling from place to place', 'fahrend' (traveling) is often used in fixed combinations, e.g. 'fahrende Händler' (traveling merchants), 'fahrendes Volk' (itinerant people), or historically 'fahrende Ritter' (wandering knights).
Früher zogen fahrende Gesellen von Stadt zu Stadt, um ihr Handwerk zu lernen.
In the past, traveling journeymen went from town to town to learn their trade.
standing
'stehend' and 'fahrend' form a clear contrast when it comes to vehicles or traffic. Here, 'stehend' means that something is not moving. 'fahrend' means that a vehicle or object is currently in motion. The difference is especially important with cars, trains, or buses. A 'stehender Zug' (stationary train) is stopped at the station, while a 'fahrender Zug' (moving train) is moving along the track.

Aus einem fahrenden Zug darf man nicht aussteigen.
You must not get off a moving train.
Der stehende Bus blockierte kurz die Einfahrt.
The stationary bus briefly blocked the entrance.