Positive the battery is good . After you get it to start it will run great but as soon as you shut it off and try to start it back up is will run for a sec then die and cough back through the intake . It ran fine never had any problems until it got cold
As Slammin Sam said, temperature effects batteries.
Car battery's use a chemical reaction to store energy, in warm temps this chemical reaction can occur quickly and release full battery AMPS when using starter motor.
As temperature drops the chemical reaction slows down, so available AMPS also goes down, the starter motor also needs more AMPS to turn colder engine, so double whammy.
After engine starts battery is not used at all, the alternator provides all vehicle power.
The Battery's only purpose is to start the engine.
Battery voltage is 12.5volts, approx.
When engine is running vehicle voltage is 13.5 to 14.5 volts so higher than battery voltage, this keeps battery charged for next time you need to start engine, but also shows you that the Battery is not used at all to power any vehicle systems while engine is running.
A 12volt battery simply can not show the 14volts you see while engine is running.
The fact the engine runs fine when started means problem occurs only when starter motor is used.
Point being, the battery can be good, brand new in fact, but the description of your problem points to a lack of AMPS when trying to start engine.
Starter motor for a V6 can draw 100amps, when you pull that many AMPS from a 12volt battery the voltage will drop, usually down to about 10volts, but this is still enough voltage for a good strong spark.
Starter motor is turning the engine but voltage is dropping to low in fuel/spark system for it to work, when you release the key(starter motor off) voltage pops back up to 12volts and you get a good spark which you feel as engine starting for a second and then back firing before dying.
i.e. when starter motor is on and if voltage drops under 9 volts the spark is very very weak, and cold engine needs strong spark to ignite fuel.
When you release the key voltage pops back up to 12volts and spark gets strong so fuel ignites, and engine seems like it wants to start but can't, it fires and then can back fire, because of the extra fuel from cranking engine with weak spark.
Adding "battery charger" gets the cranking voltage back up above 10volts and engine starts
So you need to test voltage in the system to find out why the AMPS are limited.
Starter motor pulling too many amps
Failing battery cables, corroded connections on cable ends
Battery with limited CCA(cold cranking amps)
If engine didn't start with "battery charger" connected then you would look at fuel or spark issue, but because it does then best course of action is checking electrical system during starting