Traditional Vancian casting is a pretty poor fit for this, but PSIONICS or Mana Pool variations work better for closer to narrative magic. I haven't seen Nanoha so I can't say about it, but with Psionics / Mana Pool vancian magic you can theoretically know any number of spells, but only use so much mana per cast. Core strength would be your Mana Pool that increases how many spells you can CAST, while experience would inform how many spells you KNOW.
Non device mages would be like Sorcerers, you know what you know and have to permanently swap out spells to learn more spells.
Nontemplated device mages would effectively be like
Arcanists / 5e Wizards who know a whole bunch of spells, but only have so many prepared*. Some spells could be cast with the equivalent of a 5e Wizard's Ritual, which is basically "I copied it from the textbook" and takes an impractical amount of time to be useful in combat.
Templated mages are like 3.5's Warmage, who know their whole list off the bat** but are MUCH more limited in what spells they know. Additional spells can be added via Advanced / Eclectic learning (Cross Training) but the number is very limited.
As time goes by all three types of mages learn more spells. A non-templated mage can acquire every spell, but are limited by how many they can prepare / memorize per day.
*Preparing a spell is like cramming for a test, you know it
now, but by tomorrow you'll forget about it. Pathfinder has
Spell Mastery for "I've actually learned this spell" as opposed to "I cram this spell each morning" but I'm not sure how that would apply to this story.
** Fun fact, Beguilers, Warmages, and True Necromancers are the only classes that can actually cast spells above their normal level using Versatile Spellcaster since they know their whole spell list from the start.
((
Spheres of Power is an interesting twist on Vancian casting that fits other types of narrative combat, but doesn't fit the established structure of this story as well as the above idea. It works somewhat well for a Harry Potter style caster though. ))