Output stability of step-down regulator connected to vehicle battery
Some phone charging ICs have marginal overvoltage tolerance. If your buck converter EVER exceeded the phone's spec you could cause problems. You need to be SURE both of the phone spec and the buck converter's behaviour.

Your subject line mentions regulator stability.
And, yes, some ICs in some circuits with some varying input conditions and some loads can behave somewhat differently than if input is lower voltage and steady.  Ensuring it doesn't cause problems can be addressed by   


Determining what a sensible Vsafe = V_phone_in_DC is
At least ensuring that requlator Vout does not EVER exceed Vsafe
This may be achieved by a post regulator clamp or regulator or ...
Opionally and usefully testing the regulator under a representative range of Vin and load conditions to see if it may 'shake its head and yell loudly' occasionally when suitable perturbed.


Regulator stability may vary with surges, load dumps, dips, spikes, voltages at a higher level than when engine not running etc. Sometimes maybe phase of the moon, Murphy in a playful mood and bank holidays. For the latter a voltage clamp is a useful backstop.

The issue involves BOTH what the converter does and what the phone does about it. I would definitely not rely on "no more output volts than many USB ports I've charged from" (presumably = 5.1V as noted in question) as being 'safe'. I've seen USB ports at 5.5V AND phone regulator ICs that are rated at Vin = 5.5V ABS max.  Really. That's insanity writ large, but happens. 

I do not now recall the USB spec limits but if your phone will tolerate 4.5V I'd look at setting the voltage there. 



The most important things here are probably the things that are as yet still missing.  We know that one component on the PCB is an LM2596 buck regulator. We do not have a circuit diagram or any attempt at description of circuit operation. We as yet do not know what several more ICs on the board are. Peering at blown up amazon photos - a poor substitute for the guy wity the board just reading the part numbers off, where they are intact, we can see that the 8 pin soic by the output is a TPCA8016 - an N Channel 25A, 60 V MOSFET. I'd guess it was used to implement the output on/off function which we have not been told about. 

Quite possibly pivotal is how the Vout_up and Vout_down buttons whose presence we have not been told about achieve their functionality.
Odds are they control up/down control pins on a digital pot IC. 
This MAY be the 4 pin (SOT89 3 pin + tab device) by the Vin connector block. If this is what it is then one needs to hope that it does not have flights of fancy when the Vin excerts (as auto supplies are wont to do) as if it returns semi-random values under duress Vout will be equally semi random. Not usually a good look.  

Datasheets for two versions of the LM2596 IC are available here
TI version and ON Semi version.
These are both valuable for overlapping reasons.  

One component which can have a very significant effect on regulator operation during either load or input transients is a "feed forward" capacitor across the upper resistor in the feedback divider network. This can be analysed formally in terms of frequency response but can also be understood as providing a path whereby fast output transients are coupled directly to the feedback pin rather than being divided by the feedback network and low pass filtered by the resistors plus stray and FB-pin capacitance. The effect is to cause the regulator to respond nearly instantaneously to fast transients with "fast" being determined by capacitor value and associated resistors. 

In the ON Semi data sheet in diagrams in Fig 1 on page 2-top (2 examples), Fig15, Fig21, they show such a capacitor 'CFF' connected across the top resistor in the feedback network resistive divider.
If the Chinese buck board does NOT have such a capacitor present it should have - and one should be added if absent.
In the table on page 10 at bottom they say



Feedforward Capacitor (CFF)
It provides additional stability mainly for higher input voltages. For
Cff selection use Table 1. The compensation capacitor between
0.6 nF and 40 nF is wired in parallel with the output voltage setting
resistor R2, The capacitor type can be ceramic, plastic, etc..  



See bottom of Table 1 at the top of page 11 for recommended values. They show something in the 5 nF - 20 nF range in their case. This to some extend (only) depends on the value of R1 (FB pin to ground) which can be 1K to 5K - they use 3K. What the Chinese use and if this is constant with digital pot variation depends on the circuit used - but if one end of the digipot resistor is grounded, which it more likely than not is, then R1 will vary with pot setting.  

In the TI datasheet in fig35 page 26 an equivalent Cff capacitor is shown.   



It is mentioned briefly at the top of page 16 where it misleadingly mentions use for Vout >= 10V.
It is discussed in 9.2.2.2.4 (!!!) at the bottom of page 28 where a misleading comment is again made BUT table 6 shows values for use across the whole Vout range. 

Is Cff present in the Chinese module as supplied?????????????????



The death of the cellphone battery suggests that substantial over-voltage was supplied. Use of a voltage clamp between regulator and cellphone seems liable to be a very very very good idea. 

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