Regular crashes on iPhone 4S/iOS 6.1.1: due to old battery, or because of updated-for-iOS-7 apps?

I've had additional crashes since posting my original question, and having looked at the logs I've concluded that my battery is simply on its last legs :-)

I found a number of other people online having similar problems who also concluded it was the battery, which led me to ask "What causes iPhone low battery crashes?" (aforementioned other people linked there). A helpful answer there explained that as a battery nears the end of its life, it can exhibit voltage drops (even before being empty) which result in this sort of behaviour. The logs indicate that this is what's happening.

Looking at low battery logs from previous crashes, I've noticed they've sometimes happened while I've been using stock iOS 6.1.1 apps, which rules out my idea that some updated-for-iOS-7 apps aren't fully backwards compatible and are causing this.