All inverters are modified, or rather constructed, sine wave to some degree or another.
Consider the typical UPS for computer equipment. It uses an modified sine wave inverter.
So called 'pure sine wave' inverters have become less expensive in recent years and that label usually indicates a better quality inverter but good quality MSM types do exist and are usually more efficient.
The most problematic equipment when it comes to modified sine wave inverters are some of the less expensive wall warts.
The influence of load power factor can be interesting but it is mostly an efficiency issue. Again, the typical UPS and the equipment it usually protects can provide a clue here. There aren't many power supplies that give much attention to power factor. That is why the UPS is often rated the way it is.