The battery specs are :
Model OT 9-12 ( 9Ah 12 V )
Standby Use 13.6 - 13.8 V
Cyclic Use 14.5 - 14.9 V
Initial Current less than 2.7A
( ps.. any laymans explanation of the above would be really welcome )
The 9Ah is a measure of energy, indicating (very roughly) you can draw 9A for 1h, or 4.5A for 2h, or 2.25A for 4h, etc. Though it very much depends upon how old the battery is, the history of how it was charged, temperature, actual current being drawn, etc.
Standby use rating is likely the voltage you will measure (open circuit) with no current being drawn and a fully-charged battery.
Cyclic use looks like the voltage you should charge the battery with (I haven't seen the term before).
Initial current....not entirely sure what they mean by this either but it sounds like a limitation on how much current can be used for charging.
My assumption here, is that the battery was drained down so far, that it was damaged beyond its ability to recharge ?
Good assumption. UPS systems commonly use lead-acid batteries for providing enough power to a system so it can be shut down in an orderly manner, WITHIN A FEW MINUTES! They are not designed to continuously power their loads throughout a power outage. Lead-acid batteries are intended to be kept constantly charged and start to "fail" when depth-of-discharge exceeds 50% (roughly), unless they are "deep cycle batteries" (not likely in a UPS). Basically, the more deeply you discharge it the less often it can be recharged. If you let it get 100% discharged (as it sounds may have happened) it's conceivable the battery is ruined.
You can, however, try to just recharge the battery. It's possible the PowerMax is failing simply because it can't charge a fully-discharged battery while also powering the load, without overheating. Try just plugging the PowerMax in (with the old battery) and nothing else connected to it, giving it a good 12h-24h to recharge the battery to 100%.
If this is the case, how do I prevent this from happening again ?
Exactly what you suggest
Monitor the voltage, then take action. If you notice a power failure, perform an orderly shutdown, then cut power. A voltage divider is good enough for voltage monitoring.
If you really need to be without power for 3 days you need to rethink your battery needs. A battery designed for deeper discharge (deep-discharged lead-acid, NiMH, etc.) is necessary, and likely a customized UPS system.
: compatible with Arduino UNO, 24V operation, all I/O's fused and protected