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From what I can read in the rulebook, there are three main ways to cast a spell:
Did I miss a part of the book that describes how to cast existing spells outside these 3 ways? If so, please point me to the right section of the book.
Those are the three ways, but you can use improvised spells to replicate the spells in the book by stacking spell factors or risking Paradox. The general effects listed on page 111 are just the starting point.
My criteria for inclusion in this list is whether this is 1) a way to cast a spell outside the 3 listed in the question or 2) a way to use a spell that a character does not normally have access to.
Artifacts
Though they require a mage to wield their power, these Supernal objects cast spells using their own effective Gnosis, Arcana and Mana. (p. 99)
More exotic Artifacts known as Astras, Eidoforms and Sariras are detailed in Signs of Sorcery Chapter 5.
Down and Dirty Spellcasting
When a spell needs doing but it wouldn't be interesting to give it the full rules treatment, a Storyteller can permit the player to simply roll Gnosis + Arcanum. Unlike everything else on this list, this is a purely out-of-character mechanic. (p. 117)
Grimoire
If you have a Grimoire (p. 101), you can cast its rotes directly from it without having to learn them first. This is a special case worth noting, halfway between improvisation and rote. (p. 112, 2nd paragraph under Rotes)
Hung Spell
With Time 2 and Fate 2, one mage can enchant another mage with spells that do not take effect until a condition is met. (p. 187) A mentor might use this to load their apprentice with spells beyond their pupil's grasp.
Imbued Item et al.
Any mage capable of the Imbue Item Attainment can place a spell in a mundane item. (p. 195) This grants the Imbued Item Merit which can also be purchased at character creation. (p. 102) Once Imbued, anyone holding the object (Sleepers included) can trigger the stored spell.
Signs of Sorcery Chapter 3 is devoted to magical items of various stripes, but of particular note are the spells on p. 69-70. These provide rules for storing spells in ghosts, people, creatures, spirits, places, ingestibles and even software.
Legacy Attainments
These are spell-like abilities that training, lifestyle and philosophy have embedded into the mage's soul. They often emulate an existing spell but draw upon the soul instead of the Supernal Realms and thus are completely free from the Lie (meaning no Paradox, Dissonance, Sleeper issues, etc.). (p. 198)
Supernal Summoning
Though not explicitly stated, a Supernal Being would likely have no problem replicating any single spell effect within their Arcanum. (p. 94) An Acanthus with 3 dots in Time could, theoretically, summon an Anachronism to cast Corridors of Time (p. 191) on the cabal. Clear the effect and the table-time investment with the Storyteller before you start this; Supernal Summoning is meant to be a momentous occurrence and could fuel/eat an entire session.
Tulpa
Rapt mages (called the Mad in the 2e core) that avoid acting on their Fault manifest Tulpa, subconscious spells that act on the Fault for them. (p. 236) This is explained much more fully in Night Horrors: Nameless and Accursed, particularly beginning on p. 71. Tulpa are normally not something player characters do except that the Soul Cracks Condition can cause non-Rapt mages to manifest Tulpa if they overuse a Rapt mage's soul stone. (Signs of Sorcery p. 89)
