What does it mean for a dirt bike to be in neutral? Does it mean you can push it forward or can't push it forward?

You can push it forward if it is in Neutral. It would move (especially downhill) without any gas.

You can also run the engine without the bike moving as a result, and you could push it backwards too.

When a dirt bike is in neutral, it means that the transmission is disengaged, and the bike is not in any gear. In this state, the bike is not connected to the engine power and can be easily pushed forward or backward. This can be useful when you need to maneuver the bike in tight spaces or push it around without starting the engine. Without being in gear, the bike won't move forward even if you twist the throttle or release the clutch. However, it's important to note that being in neutral doesn't mean the engine is turned off. The engine is still running, but the power is not being transmitted to the wheels.