MacBook Air's battery doesn't charge, Service Battery state, but only 400 cycles
It happens to me on my MBA, no charging light on. 

By moving the mag connector (like let it snap in few times) it then comes up.

Just a bad connection. You can try cleaning it with alcohol.

And I also have the "Service battery" but that is only a recommendation.

Next level would be displaying "Replace soon, or Replace Now.

While the light on the connector is not on, I can see in the battery indicator it is still charging, telling me the wall plug adapter/converter is still working.

To check the health status of your battery, look in the about this Mac - Power, 

Charge Information:

Charge Remaining (mAh):   2712

Fully Charged:    No

Charging: No

Full Charge Capacity (mAh):   5014

Health Information:

Cycle Count:  528

The Full Charge Capacity will tell you the health status  >5000 or higher is good.