What does an Apple "battery service" involve?
That battery, while not old from age, has seen some, and then some more.

The charge count shows it as half life, but the charge it can hold shows it as 1/3 of life left.

The Service Battery is only a suggestion so do not wary about that one. The Replace battery is the next you get, and then the Replace Now is the final stage.

Considering the Charge it can hold iI would suggest you replace it. But yu can try the SMC reset to see if that fixes it. One full discharge (let the computer run till it shuts down due to empty battery) could also help in calibrating the numbers it reports.

For comparison here is my MacBook Air 2012.

Charge Information:

Charge Remaining (mAh):   3967

Fully Charged:    No

Charging: No

Full Charge Capacity (mAh):   4801

Health Information:

Cycle Count:  619

Condition:    Service Battery

Battery Installed:    Yes

Amperage (mA):    -1132

Voltage (mV): 7933