How do I create a custom cantrip for prepared spellcasters?

[Hydrothermal]

I created a spell by copying an existing cantrip (acid splash) but it doesn't appear in the list of available spells when I add a level of Wizard and edit the spellbook. I can see it in the Sorcerer list if I add a level of Sorcerer, but it still doesn't show up in the spellbook list. It does work if I change the spell's level to 1. Any thoughts?

Relatedly, is it possible to add to the list of free spells that a Wizard (or other spellbook user) starts with, without copying and overriding the class?