Great discussion! Inverter efficiency is calculated as loss of power of input to output. Since DC is converted to AC, (DC to RMS) the constant is power in watts. (P = I x E) In the discussion above, the efficiency is generally referring to the system efficiency, which takes into consideration loss through the distribution. These are two different efficiency discussions. A pure sine wave inverter is more efficient than a modified inverter, simply because you get more power out through the conversion. (Typically, 90% vs. as low as 70% on a modified sine wave.) The modified inverter has less power out for the same current in, as described above.
So, to understand system efficiency, wire size comes into play here. The more power loss you have through the wires, the less power you have at the inverter input. The efficiency of the inverter is the same in any case. But with lesser gauge wire, you now have less power at the input, and the inverter will work harder with lower voltage to attain the desired output. Heat is the result of all this extra work. (And generally buzzing) Heat is loss. It just goes downhill from there.
A pure sinewave inverter may require more power at the input to maintain its output and efficiency, therefore, the wire size has to be big enough to overcome any loss that would be the case with smaller wire. Bigger is better.
@BroncoHooves has an ideal system that if you look at his avatar, shows the drop
at the inverter, is minimal. This allows the inverter to operate at its full potential without much loss in the system delivery. This may be overkill for a 400-watt inverter conversion but drives the point.
So, what's the problem with modified inverters? Simple. Many devices with transformer inputs (typically any device that plugs into a wall) require a constant change (RMS sinewave) to meet the maximum efficiency of the conversion for input to output for the load. (Device conversion, not the vehicle inverter) Transformers need constant change to pass constant output. A modified inverter is stepped, and depending on how many steps are in the design, the "change" is only realized at the moment of the step. The more steps, the better the conversion. This is a very inefficient transfer of power. This is why many devices will work in the home, but not in the vehicle with the modified inverter. Some devices based on design, will work "OK", but not to full efficiency.
Hope this helps.
Here is a link describing inverter efficiency.
www.inverter.com