That is how the US military accounts for them, yes. The engine is not included in the price paid to Lockheed Martin.

Oh interesting. Can they get them with a different engine? Like an add-on when buying your car. Maybe like a sport mode?

At the time they were the best production engines in the world. General Electric originally was competing with the Pratt and Whitney engine F-135, called the F-136 but that was since canceled. However they're is the GE Adaptive Cycle Engine that has since been developed which produces greater thrust while having the ability to also achieve greater fuel efficiency for greater range by changing it's characteristics.