All of that, but, mainly what you're overlooking is that the "next" spell isn't always strictly speaking the "most powerful" spell. For your BLM example, Fire 2 is an AoE spell. Fire 1 is not. Fire 1 is the stronger spell, unless you are facing a larger group of enemies, when Fire 2 will be more effective. Fire 3 is also single target, and may seem stronger than Fire 1, but factoring in time to cast as well as MP cost, Fire 1 still remains the optimal choice.

With any of these skills, it's about understanding where they fit into your kit, not necessarily just using the newest one. For skills that actually get upgraded, you will find the skill itself on your hotbar with its icon will automatically change to the strongest version available when appropriate. When you get multiple different skills like you have, it's typically because each has its own different time and place to use.