I'm sicilian myself (living in Mazara del Vallo and studying in Palermo) and I make arancine with my mom quite often. I can assure you that the batter is the traditional way.
As of the difference between batter and egg, I can't really tell because we always use batter. Probably it lacks of that eggy flavour, which is good since you don't want that.

I wanted to point out a thing of GialloZafferano's recipe. They use caciocavallo cheese both for rice and ragù filling. It's a very specific cheese from south Italy and it could be hard to find outside. You can switch that with regular parmesan-like cheese for rice and mozzarella for fillings.
Also there are just two different fillings in that recipe, but you can use whatever ingredient you like. Popular ones are: speck and fontina/provola, spinach bacon and mozzarella, eggplants tomato sauce and pecorino, salmon and besciamella.
You can even go more gourmet if you like with fancy combinations.

Those are good looking arancine BUT that recipe you shared isn't very traditional (no saffron on rice, no batter and sauce on plate?).
Try this one next time, it comes from the most popular home-cooking website in Italy.
It has hundreds of traditional italian recipes and it's absolutely trustworthy.
Arancini - Italian recipes by GialloZafferano