The genuine Pro Mini's use a MIC5205 regulator which should accept up to 16V at it's input normally, with an absolute maximum of 20V. It's unlikely that the regulator would be damaged by 15.1V.
However, the component you have indicated that has blown is a capacitor. SMD capacitors are available in different voltage ratings, typically 4V, 6.3V, 10V, 16V, 25V, 35V and 50V (and above, but let's ignore them for this). It's rare to see any intermediate values.
SMD capacitors are very intolerant of being used at a higher voltage than specified. This is especially true of electrolytic and tantalum capacitors. The blown capacitor on the genuine Pro Mini's is polarised (the grey band not he package and the schematic indicates this), so we can infer it is either electrolytic or tantalum. A 10µF SMD electrolytic is unlikely to be in this package, so it is almost certainly a tantalum.
Sparkfun sell 10µF tantalums, and they are rated at 16V. It's quite likely these are the same ones used on the Pro Mini. 15.1V is very close to 16V - in fact, if this is a poorly smoothed power supply, the meter might show 15.1V but the peaks could well be 16V or even 20V. It is recommended that you de-rate the voltage ratings on tantalums by 50% at least.
It's really important to realise that most low-cost DC power supplies are not regulated and produce a very bumpy output voltage with no regulation:
Tantalums also have no tolerance for reverse voltage, so if you did apply reverse voltage, it would pop.
So it is quite likely that you over-volted the cap and blew it. You could replace them with higher rated capacitors - I would go for 35V ones if you want to supply 15.1V. Desoldering and desoldering a single capacitor is relatively easy as long as the pads haven't been damaged. Realistically, you could use a normal leaded 10uF electrolytic between RAW and GND instead of the SMD one.
EDIT: However, looking at the data sheet for the regulator, it shouldn't really matter if that 10uF is there or not:
A 1μF capacitor should be placed from IN to GND if there is more than
10 inches of wire between the input and the ac filter capacitor or if
a battery is used as the input.
Some regulators absolutely require an input capacitor, with others it is just advisable. This looks like it is just advisable. So if it isn't working on external power now, you may have blown the regulator as well.
Even if the clone board uses a different SMD regulator, their characteristics are all very similar. As stated, some do need a capacitor on the input to work, so replacing it might fix it.
There is a further consideration that dropping from 15.1V to 3.3V on a SMD regulator is not a great idea. You will need to burn off a lot of power.
The MC5205 can dissipate 455mW with a minimal PCB footprint (which the Pro Mini has) with a 25°C ambient:
So, we know the maximum power we can dissipate, and we know the voltage drop - we can then calculate the current:
P = (Vin - Vout) * I
0.455 / (15.1 - 3.3) = I
I = 38.5mA
This isn't very much at all. You might want to consider regulating off the board, and probably with a switching regulator rather than linear regulator.
I also note your PS:
the ground of the 12V PSU is shared with Arduino ground pin, not sure
if this might be the cause.
If they weren't shared, there would be no way that this could work. They need to be shared otherwise the Vcc from the power supply is not referenced to anything and may as well not be there.