Am I correct that adapter mode would burn a US appliance, or am I wrong?
Yes, you are correct. If used in "adapter" mode, the voltage is not adjusted or converted in any way, and in most cases a device expecting only 120V (USA/North America/etc.) will be damaged when plugged into the 240V source.
But it's worse than that. Even when used in "converter" mode, you may still damage a device expecting 120V. At the very least, most such devices simply won't work. Only simple loads insensitive to the exact nature of the electrical waveform will work (e.g. hair dryers, curling irons, etc.), and other types of devices may still be damaged. Read on for additional details concerning that risk.
all of those I've seen are both Converters and Adapters. As far as I know, that means that if I accidentally flip the switch to adapter mode and then connect my US appliance in Europe - it'll burn.
I felt suspicious because there is no way to fit a 2000W converter in this thing. Turns out I was right, there is no real converter inside...
Please read the fine print on the side of the "adapter/converter" you linked in your question:
So it is only suitable for resistive loads, like heating elements and of course hair dryers.
That's because the cheap "converters" you've been looking at contain a circuit equivalent to a lamp triac dimmer. This does not output a proper 110 Volts AC! It outputs the original 230V mains voltage, chopped up so that on average it heats up a resistor the same as 110V AC, like so:
But that is only suitable for resistive loads... Not for anything else, as it has peaks higher than normal 110V mains peak voltage so it will probably destroy any other kind of equipment rated only for 110V. If the load is a switching supply without power factor correction (ie, any small charger) then even if it tolerates the extra voltage, the input smoothing capacitor won't like the high current spikes, which may fry it and/or the "converter" 👍
TL/DR: don't buy that!
If you want a proper 230V to 110V AC converter, it will be a bit bigger and much lower power (this one does about 200W) ... because it actually contains a circuit that converts the voltage ...