You seem to discount a third scenario: building a larger ring of walls (or an extension) while the existing wall are still standing. In my own area of interest, Italian cities, there are many cases where an inner ring of old walls is contained within a newer ring (Milan, Verona, Bologna, and Florence all exhibit this). Of course, there could be differences across different time period and geographic areas.

I actually wrote an older answer about the demolition (and construction!) of urban fortifications which you might be interested in. I also wrote this shorter answer on city walls and Italian urbanism that you might also be interested in.