I and several others have already answered each of your questions several times in different threads, responding directly to your posts. If you don't "get it" by now, maybe you might try going back and reading the responses again. I'll save you the embarrassment of quoting every time you asked and every time someone answered your question, and you still didn't "get it". If you feel condenscended, you surely deserve it. Now that I have your emotional attention, take a deep breath, because here are the answers. Again.
Batteries die from neglect. (Improper electrolyte level resulting in sulfation, allowed to excessively self-discharge resulting in sulfation, failure to keep connections clean resulting in heat build-up due to increased electrical resistance, etc.)
Batteries die from heat. (>85*F is tough on batteries, whether ambient temperature >85*F, or the battery temp exceeds 85*F from exothermic chemical reactions due too excessive charge or discharge rates and/or electrical resistance due to excessive charge or discharge rates.}
Batteries die from freezing. (<32*F is tough on batteries.)
A lead-acid battery self-discharges at a rate of 8-40% per month, depending on how much antimony is alloyed with lead in the plates. Alimony is alloyed with lead to provide tougher plates less prone to vibration and impact damage. Motorcycle batteries are subject to severe vibration and impact loads, so their plates contain high antimony levelos, so their self-discharge rate is closer to the 40% than the 8%.
Lead-acid batteries loose capacity as temperature drops below 32*F. For example, the battery in my Olds is rated at 750 cold cranking amps at 32*F. At 0*F, the same battery is rated at 600 cold cranking amps. That's a 20% drop in capacity. So, a battery that sits uncharged and self-discharges 40%, plus is freezing at 0*F and looses another 20% of capacity, only has 40% of capacity remaining. Not even a deep cycle battery should be discharged below 40% of its capacity. Starting batteries should never be discharged below 10% of their rated capacity.
Now, a battery discharged to 60% of its capacity, and someone hits the starter, and is surprised when the battery is damaged by being asked to work at 250% of its practical capacity?
In your case, with the charger attached, if stored much under 20*F, your battery was being maintained at 90% of its capacity after it was charged. Leaving the charger on all the time probably slowly sulfated the battery, which results in degradation called "surface charge", which will often start a vehicle if it starts right up, but lacks depth if starting requires any significant cranking. Funny thing is, a surface charge will test normal voltage when the battery is not being used, but the battery voltage can drop below 10 volts within seconds under a load. No voltage, no amps. I expect the constant charging, at too high a rate, and cold storage, combined to cause your battery to sulfate. I wouldn't be surprised if your charging circuit is just fine, it just doesn't put out enough voltage to power through the sulfation to force a surface charge on your battery.
Tony, you're output is 15 volts and 600 milliamps. DC. I expect the AC claim is bogus. If not, your charger will not charger your battery, except the little square box between the wall wart and clips is packed with diodes that convert AC to pulsed DC. Also, the voltage is reduced to 13.2 volts due to resistance in the wires and electronics, which is a 12% voltage drop, which would likely provide a 12% drop in the rated 600 milliamps, resulting in a practical output of 528 milliamps, which is a bit high. I think you get away with leaving it hooked up because the pulsed DC voltage is relatively low, at 13.2 volts. A fully charged 12 volt battery in good condition will actually measure between 13 and 14 volts. As the battery voltage approaches the charger voltage, the current (milliamps) slows, and when there is no difference in volts, there is no current flow. In essence, it is the low output volt that prevents battery sulfation.
Also, heat is the enemy of a battery. Your slightly high milliamp rating would easily overheat the battery if used to charge in ambient temps close to 85*F. I expect you don't let the battery become excessively discharged where the charger would take several hours to restore the charge, so even though the charger is hooked up and plugged in, the actual charge time is too short to heat the battery very much. If you use this charger in the summer, it might be a good idea to use a lamp timer with 15-minute switches, so the charger cycles on 15 minutes, then off 15 minutes, which provides time for the battery to cool between charge cycles. In fact, a 40% self-discharge rate would easily be restored by about 12 hours use of your charger per month, assuming the typical 50% efficiency, with 50% lost to internal resistance and heat. Your battery would stay fully charge if you ran the charger 30 minutes a day. During the summer, use a timer to due a charge cycle about 6:00AM, when ambient temps are coolest. Unless you store the battery in an air conditioned structure--then it doesn't matter.
Note that the instructions state that your charger is not to be used to charge a battery. Well, considering the 125 amp battery maximum would take about 600 hours to charge, that might be good advice. More than likely the charge rate would be lower than the self-discharge rate of such a large battery. A 125 amp battery would be a starter battery of about 1250 cold cranking amps, which would be about right for a diesel pickup. Be careful using it in the summer, all well be well.