I think the idea is that firebending is less literal fire and more energy source which would include eletricity but energysourcebending doesn't roll off the tongue as well as firebending

Fire is plasma and Lightning is higher form of it. I think I wrote there are a lot of other forms of bending in ATLA and LOK. In this season, they deep more the concept. Anyone in the Fire Nation(who are good enough) can access to Ligthningbending but the royal family keep it from themselves.

Azula is definitely a nice addition to the series.

To every bending form there are sub-bending techniques. Usually a small number of benders is able to learn them, because of skills, talent, the right mindset or something like that. You've already seen healing as a sub-bending technique of waterbending. Now lightning is a technique of firebending. (It will explained in a later episode how it exactly works. It's actually real life physics adjacent. 🙂) There are several other sub-bending techniques that will be introduced in The Last Airbender (and its spinoff show Legend of Korra).

She was sad cause they were stealing their osrtich horse after letting them into their home.

Lightning will be explained in a few episodes, this show rarely leaves any loose ends

In regards to Iroh's intervention of Azula's lightning attack, many seem to think Iroh redirects it. But to me, it feels more like a physical push of her arms to another direction before the decharge so it misses the target.