The first word that comes to mind is schism .
According to the American Heritage® Dictionary of the English Language schism : A separation or division into factions: "[He] found it increasingly difficult to maintain party unity in the face of ideological schism over civil rights" (Nick Kotz).
Obviously, there are many synonyms listed in the dictionaries that can be used.
ell.stackexchange.com/users/3223/Lucian Sava
As I was typing this question, I suddenly remembered the word divergence .
The situation in which two things become different.
Cambridge Dictionary
I'm posting this answer for peer review, since I'm not sure.
Also, I'm interested in other options, too.
After the divergence of Western philosophy into these major directions, new thinkers began to emerge.
What is the historical/etymological explanation for this divergence in names?
ell.stackexchange.com/users/26160/Revetahw says Reinstate Monica
bifurcation the division of something into two branches or parts.
After the bifurcation of Western philosophy into these major directions, new thinkers began to emerge.
What is the historical/etymological explanation for this bifurcation in names?
ell.stackexchange.com/users/5766/Dean MacGregor
For the first example, you might be interested in: division : the act or process of dividing something into parts : the way that something is divided After the division of Western philosophy into these major directions, new thinkers began to emerge.
branch : to divide into smaller parts : to separate into branches After the branching of Western philosophy into these major directions, new thinkers began to emerge.
After Western philosophy branched into these major directions, new thinkers began to emerge.
separation : the act or process of separating : the state of being separated
After the separation of Western philosophy into these major directions, new thinkers began to emerge.
Similar for the second example.
ell.stackexchange.com/users/36187/Em.
You could consider Fork From Dictionary.com : the point or part at which a thing, as a river or a road, divides into branches
ell.stackexchange.com/users/4995/Tom Bowen
The first word which came to my mind is fission .
It's perhaps most commonly used in the context of nuclear physics, but it doesn't need to be restricted to that context.
It could go directly into your first example sentence: After the fission of Western philosophy into these major directions, new thinkers began to emerge.
Looking through Google Books, I get the impression that it's a moderately popular word in the social sciences.
E.g. in Encounters and Transformations: The Archaeology of Iberia in Transition
we find sentences like ...
incipient complexity led more often to social fragmentation or fission than to pristine state formation.
I argue that too little attention has been devoted to fragmentation and fission and that more sophisticated models be developed to account for these alternate trajectories of social evolution.
Google Books also turned up the word in a number of titles or subtitles, often in opposition to fusion , as in the white paper Fission Or Fusion: What Kind of Commercial Culture Will Emerge in Southeast Asia? , but not always.
An example without fusion is the book Household Strategies for Survival 1600-2000: Fission, Faction and Cooperation .
The British National Corpus turns up a couple of nice examples in natural science literature.
From The Pacific by Simon Winchester: The slow fission of Gondwanaland produced two oceans -- the Indian, where Africa, India and Antarctica were hauled away from each other; and the southern portion of the Atlantic, where South America and its clearly closely-fitting neighbour Africa (which possessed an uncanny coastal match first noted by Francis Bacon) sprang apart.
And from Exploring the night sky with binoculars , by the legendary Patrick Moore: It used to be thought that a binary [star] resulted from the fission or breaking-up of a formerly single star which was spinning rapidly, and became unstable.
ell.stackexchange.com/users/12483/Peter Taylor
Bifurcation Meaning : to branch out in two For example: The path bifurcated on reaching the forest.
Ramification
Meaning : diverging branches or consequences For example: The army killed all the terrorists and all the related ramifications.
Fissuring
Meaning: branching , splitting For example:
The heavy earthquake created apparent fissures on the playground.
ell.stackexchange.com/users/40755/aelin_s
If one can overlook the second meaning in the Oxford Dictionary , the first definition of cleavage is A sharp division; a split After the cleavage of Western philosophy into these major directions, new thinkers began to emerge.
What is the historical/etymological explanation for this cleavage in names?
Here are some other examples from the Oxford Dictionary:
There was not this sharp consciousness of the cleavages and different realities in our social existence long ago as that which is so evident today.
To existing social splits were added inter-working class cleavages which were all too apparent by November 1916.
It is a function of that same openness that no sharp cleavages can be sighted between the traditional and the modern in India.
ell.stackexchange.com/users/30978/JavaLatte
Perhaps a couple less literal but more literary options would be the nouns "cleft" and "rift."
Verbs would be "to cleave" and "to rive."
The issue created a rift in the party.
The setting sun shone through a cleft in the ridgeline.
ell.stackexchange.com/users/40753/Charlie Tango
I know there's a perfect word for this, but I just can't seem to recall it.  I know about  split , but I'm thinking of something more eloquent.  For example:   After the  split  of Western philosophy into these major directions,  new thinkers began to emerge.   For another example, consider this sentence from  this question :   What is the historical/etymological explanation for this  split  in  names?
Noun for the event of something splitting in two directions
ell.stackexchange.com
<word-request><nouns>
ell.stackexchange.com/questions/101518
ell.stackexchange.com/users/26160/Revetahw says Reinstate Monica
Suggestions include schism, divergence, bifurcation, division, branch, separation, fork, fission, cleavage, cleft or rift. "Schism", "Divergence", "Bifurication", "Division" (division into parts), "Branch", (as in separate into branches), "Separation", "Fork" (used to indicate the point where something is split) and "Fission" could be used instead. "Cleavage" can also be used. Finally, "Cleft" or "rift" are some less literal synonyms.
"Schism" is a synonym. "Divergence" can be used. "Bifurication", which means division into parts. For the first example, "division" could work. "Branch", as in separate into branches. "Separation" is an alternatibve. "Fork" is also used to indicate the point where something is split. "Fission" is another option. "Cleavage" can also be used. "Cleft" or "rift" are some less literal synonyms.