It sounds like you've found the solution. However...
My ex-gf had starting problems with a second-hand car. We'd bought a new battery for the car the day we got it, so it couldn't be that. After some questioning, it turns out she thought she'd left an interior light on.
She didn't understand why this would flatten a new battery, because interior lights don't take much power. I explained that batteries don't necessarily come fully charged. She might have got away with it with a fully charged battery, but as it came off the shelf it wouldn't.
So that explained the initially flat battery. A jumpstart got the car running, but then she couldn't understand why she'd run the car for half an hour, and it would restart after turning it off but not a few days later.
The problem simply was that the battery was so flat that half an hour with the engine running wasn't enough. The alternator put enough juice back into the battery that it could handle one round of starting the engine. But a second round, with a cold engine, and with several days of the battery drifting back down, was more than it could handle. Half an hour of alternator was the equivalent of pouring a cupful of fuel in an empty tank, when it actually needed gallons.
The solution was that I disconnected the battery, put a mains-powered battery charger on it, and left it for 24 hours. Car starter batteries aren't designed for deep discharge, so I was worried that she'd killed it. But I connected it all back up and it worked perfectly, and it's carried on working ever since.