Financial stability just means I have to respect my financial limits, never spending money I don't have or that I will miss later for emergencies.
Your definition of financial stability sounds like you are emphasizing about the importance of saving money for better future. Different people must be having different opinion and hence I just agree with you because without having enough money, we cannot think about a stable lifestyle so saving is more important to think about financial stability.
Always having some income stream to lead our life stable could be defining what I feel financial stability should be like.
Yes, to not be able to save some money at the end of the month means the person doesn't have financial stability, so savings have a very important role in an organized financial life, that is the key for improving life quality while being responsible at same time. Maybe in some cases the individual can manage to have stability, but spending everything he earns, what means he doesn't save anything. The problem with this approach is that he won't be able to fulfill any long term goals in his life. It will be eternally "tied" without any development.