I don't really understand the way lighting inverters are rated either. Look at page 4. These small lighting inverters are rated the same VA and W, yet they specify a PF rating of 0.44. These two conditions can not be met simultaneously. The LPS26, for example is 20W/20VA with PF rating 0.44. This power factor rating is consistent with straight forward switch mode power supply input that uses a bridge + capacitor. A simple old school mass market corkscrew 18W CFL is probably 18W/36VA with ~0.5PF.
My guess would be that real power is limited to 20W, but the back-end of inverter is built with enough margins to handle the poor power factor reasonably anticipated small load, for example, using the lighting inverter to fire up one off the shelf 18W corkscrew CFL as emergency fixture. By not putting 20W/45VA, they avoid unnecessary support calls about this.
Their larger units are rated at PF=0.8.
1kW/2kVA is practically the same thing as 1kW with PF rating of 0.5. If we're talking about a genset, the VA determines the alternator sizing, but it's the real power that determines the load on the prime mover, thus the fuel consumption. Similarly, it's the real power that matters when it concerns the runtime.