Warning light flashing points to worn brushes on the alternator, if stop/start then stop now, as you need a computer to work out what is wrong. But if not then engine running slightly over tick over and should show 13.4 to 14.4 volts, yes seems a big gap, but some alternators have stage charging, but normal is 13.8 volt.
We had a problem selling sealed batteries, we had to charge them before they would be replaced, and even then we could not see the electrolyte, if over charged it goes reddish colour, so once returned to manufacturer they would soon know if failure down to abuse, and so we could end up replacing a battery and then being refused replacement by manufacturer, so if under 12.4 volt then we assumed abuse, and would only replace if manufacturer refunded us. I think that was wrong, but I was not the boss.
Manchester batteries would always replace, now Tayna, so we would always replace to customer, more well known makes we had problems with, it seems motorist shops had been replacing batteries without question, so the manufacturers tightened down.
The warranty at that time required us to fit the battery and enter vehicle details, taxi's only had 6 months warranty, not a clue why, so if we had fitted it, we should have checked charging, but we did sell without fitting so did not always check charging, today with stop/start not sure how shops could check charging? It needs a PC plugging in.
Off car charging needs to be with a smart charger, and most need at least 8 volt to start charging a 12 volt battery, under that it assumes a 6 volt battery is being charged, so with the smart charger I have had to cheat, and connect two batteries in parallel to get charger to work.
My Honda Jazz had a battery which would only last 3 weeks without use, it was clearly faulty for last two years, in September it decided to fail completely, but I had been given loads of warning. And normally you do get some warning battery on the way out, not the same with alternator, they tend to work one day and not the next.
Also we get the lazy starter, in the main that means bearing have warn, so if engine warm and easy to turn over works fine, but if engine is cold then the extra power causes the armature to bend slightly and catch on the pole shoes, it can be miss leading as often a jump start will turn engine over OK, so people assume battery US but really it is the starter.