Everything stills point towards a faulty battery. It's quite possible that the shop got a bad batch of batteries or, I hate to say it but, perhaps they never replaced it the second time seeing as how they seemed unconvinced when you brought it back. At any rate a large battery like those in the iPads can supply small amounts of current for an incredibly long time, even a very weak battery. However, when the current demand increases, the battery life gets reduced significantly. Under normal conditions, you should get ~8-10 hours or normal use and days worth of standby time.
What I suspect is happening here is that when you power up the device for activities, the battery is simply degraded and shutting down completely. To be certain, use a battery utility, such as [http://www.coconut-flavour.com/coconutbattery/|coconutBattery] (for Mac) or [http://www.3u.com/|3uTools] (for Windows). It will tell you what the health of the battery is. Anything less than 70% of design capacity will require replacement (and should normally be covered by the shop)