Shouldn't I only be able to add at most 2 level two spells to the spellbook? I guess that is how I prepare the spells, right? Or did I understand something wrong here...?
That's right, it's the difference between Prepared spells and spell slots.
When you select spells in the spellbook, you are picking which spells Shadowheart has Prepared. This provides you a list of spells that you can use in combat as you choose. You might choose to prepare more spells than you have spell slots in a certain level, because it gives you more options to use.
Older editions made you choose ahead of time what spell goes into each spell slot, but 5e gives you more flexibility. When you choose to cast a spell, it can be any spell you have prepared, as long as you have an appropriate spell slot to use to cast it.
It's a good idea to prepare spells that you can cast in lower level slots as well. Spells can be "upcast," which means using a higher level spell slot than it needs (so you can cast a Level 1 spell with a Level 2 spell slot), but they cannot be cast with lower level spell slots (a Level 2 spell needs a Level 2 or higher spell slot).
