Some champions, like Sivir, I see recommended to start with Chalice of Harmony and a lot of other champs is see recommended to start with Catalyst the Protector. I'm wondering what characteristics of a champion to look for that determines a preferece of one over the other. What factors should I consider when choosing between the two?

I prefer catalyst of the protector until around level 15, as the boost from leveling up keeps you in lane when you are low leveled and it is very good for facing caster heavy teams because it can build into banshee's veil later.

Catalyst the Protector is an item that's primarily used to help you stay in your lane.

A character who can benefit from Catalyst is one who...

1) Can benefit from Sapphire or Ruby Crystal as their first item.
2) Has a hard time staying in the lane, usually squishier melee characters
3) This isn't 100% important, but they should be able to benefit from Rod of Ages, Banshee's Veil, or Innervating Locket later on.

Chalice is not quite designed that way, but does pretty much solve mana problems in the early and mid game.

1) Good for a character with a heal or some kind of spammy defensive ability
2) Good for a character who doesn't have a hard time staying in the lane
3) Good for a character who harasses with abilities

So, a Chalice is very good for Sivir if you rush it; you can spam her abilities and she's ranged, so she doesn't usually get hurt during the laning phase.