To me, the two definitions are just rewording of each other: not two different definitions. If event 1 happens during the time that event 2 happens, event 1 happens at the same time as event 2.
Let's look at "as" and "while" as synonyms. I would use "as" instead of "while" only if the stress was on the duration of time (as opposed to location or state).
I worked for a radio station while I was at university.
Here, I worked at a radio station at some point during my tenure at university, not during the entire duration. The time at university is used more as a state and not as a quantification of time. The whole time spent at university is not the focus here, just the fact that I was at university.
I listened to the radio while/as I waited in the car.
Here, we are trying to stress on the time spent. We want to elaborate on how that time was spent. We are using "I waited in the car" as a duration.
Typically, if I am using a clause to describe the duration of an action, I would use either "as" or "while": while/as I sat, while/as I spoke, while/as I lay, while/as we talked, etc. If I am trying to describe a state, instead of focusing on the duration, I would use "while", not "as": while I lived, while we were at the park, etc.