Though I'm with you on the pure sine wave thing, I have found that the e bike chargers that are my load, care not. The crappiest Harbor Freight square wave inverter, (bought in a pinch, to my surprise it's still kicking a coule years later) and my MorningStar 300 watt inverter (a very respected name in the alt. energy world) pure sine wave seem to both get the job done. It's hit or miss, as to which type, depending on the type of load of course, when in doubt just go pure sine wave for sure.
If you ever find a type of load that does not like sine waves, that'll be news. Given that the once-large price premium for sine inverters is now small, I just don't feel much like experimenting to see which of my loads tolerate anything else. I have a few old MSW inverters sitting around that I no longer use, but haven't taken to Goodwill either, mostly because in this day and age I don't really want anyone else ending up unknowingly using them either. I suppose I'll get them to the electronics recycler one day.
Just note.. a sine wave is what equipment is designed for. Super-impose a mod-sinewave on top of a sine wave and most everything outside the sine wave is wasted as heat and-or vibration. Your call.
