If you're going to buy one anyway, get one that can do a "sample and hold" of what is called "inrush current". Very handy for reading the momentary spikes when loads start up, and since they happen so fast, you need a meter that can not only read it, but can remember it.
Huh. That's whack. Does it heat the water? Maybe the heating element wasn't going?.
Truth is, there's no such thing as modified sine wave, that's just marketing BS. There is sine wave, square wave and modified square wave (steps added to the square wave to more closely resemble a sine wave). If it was a sine wave to begin with, they wouldn't have to modify it. Resistive heating elements are immune, they don't care either way, but I have no idea what other electrogadgetry is inside that 'spresso machine. 99% of electrostuff works just fine on MSW (including so-called "sensitive" electronics, which usually have their own power supplies (bricks) and the power supplies don't care). The problem is, there's no hard and fast rule about what will work, so you have to try every particular device to see if it will work, And if it doesn't, it's usually fried immediately. Most common fails I think are battery chargers for power tools, camera batteries and suchlike.
Before I'd suspect the wave form, I'd really want to make sure the voltage and frequency were within spec - and stays within spec when servicing a load.