Take a look at the spec sheet for your water boiler, what does it says its max possible draw is. Just because the device pulled 12.9 when you measured it, doesn’t mean that is the max it will ever draw.
Was this the first run or has it been running for awhile?
With all due respect, ‘what went wrong’ were the assumptions you made. You assumed a device rated to handle 15A @120VAC is adequate for handling a steady current of ~13A.
Here’s what you overlooked:
This device is self-certified and is never tested by an independent testing authority like UL, ETL , CSA, etc.
It’s allegedly rated to handle 15A @120VAC but (evident from the photos) has no overcurrent/overvoltage or thermal runway protection. In other words, it has no means of protecting itself from catastrophic failure (i.e its only “protection” is to melt).
You were planning to run this device at 87% of its rated current-handling capacity without consideration of the accompanying increase in temperature and how this device would handle it (i.e. poorly). In other words you did not “derate” the device based on the expected load conditions.
There’s also the matter of inrush current. That’s the initial brief inrush of current drawn by the load when first turned on.
tl;dr
This device is under-designed and inappropriate for switching a high-current load.
My personal view to your case is that your contact made between the Sonoff and power cable was not good enough to handle 13Amps, its getting hotter and hotter until the plastic connectors melted.
There is no sign of defects on the Sonoff PCB, meaning its able to handle your 13A resistor.
Weakest points broken first, so next time measure the current, plus the temperature of the connection to be sure.
To be honest, with such high currents I wouldn’t trust any device that is not certified by a well known authority. Spending double or triple the amount of money on a certified device is well worth the investment if the cheap one burns down your house.
To be honest I would always consider a Shelly over a Sonoff, it’s certified, you’re supporting a EU company and it has great customer support.