I think it is a mistake to correlate the analogy of the chariot and two horses exactly with the Republic's tripartition of the soul. In the Phaedrus:
All three capacities, and
not only a rational part of the soul, are given an essential and positive
role in striving toward the good and the beautiful, and each capacity is
represented as having certain defects. Although the three capacities have
some similarity to the three parts of the soul in Republic 4 - reason, appetite, and spirit - they should not simply be equated with these parts,
for all three capacities share to some extent in reason and all three have
desires. The charioteer represents a guiding principle in the soul, with
desires of its own. The black horse represents an impulse to move in
bold and disorderly fashion toward erotic objects, while the white horse
represents the impulse to stand still and to resist these objects. Both
horses are able to use and to follow reason and are therefore capable
of being trained by the charioteer, who must also train himself to guide them without imposing excessive restraint or yielding to the impulse to
move forward without any restraint. In this myth, all three capacities of
the human soul share in a divine, winged element, and all three also have
bestial characteristics. The black horse, then, is not innately evil but can
be a force for good if he is properly trained. Without the guidance of the
charioteer, the black horse moves in shameless and disorderly fashion,
bending his head, stretching out his tail, taking the bit in his teeth, and
dragging the chariot shamelessly forward in pursuit of an erotic object
(254d6-7). After he is tamed, however, the black horse helps to move
the chariot in an orderly way toward the beloved, that is, with reverence
and fear (...254e9).
(Elizabeth Belfiore, 'Dancing with the Gods: The Myth of the Chariot in Plato's "Phaedrus"', The American Journal of Philology, Vol. 127, No. 2 (Summer, 2006), pp. 185-217: 190-1.)
In the Phaedrus all three capacities come into agreement with one another, with the result that there is order in the soul, but this occurs only after a rough journey throughout which the soul lacks all stability. Key point: the three capacities operate in the charioteer and the two horses: 'thymos (θυμοειδές) or thumetikon' invades the charioteer and both horses - all three players - at different stages of the journey. Specifically and en route, it cannot be 'located' in either horse. The following extract is long but the story can't be shortened:
The process begins when the charioteer, seeing the beloved, warms
the whole soul, causing it to be filled with tickling and desire (253e5-
254a1). Although the stimulus comes first to the charioteer, all three
parts of the soul have the same emotional response, characterized
earlier as a boiling and tickling (251c4-5), resulting from the growth
of the feathers, that affects the entire soul (251b6-7).
The three parts of the soul act differently in response to the same
emotional stimulus. The white horse "compelled then and always by
aidos, restrains himself from leaping upon the beloved" (254al-3).
The black horse, however, is himself carried away by force and in
turn compels (254a5, bl) the white horse and the charioteer to approach the beloved and to mention the pleasures of sex (254a3-7).
They at first resist (254a7-bl). Finally, however, the white horse and
charioteer yield and agree to do what the black horse orders, and
they approach the beloved (254b1-4).
When they are forced to draw near to the beloved, the charioteer
sees his beauty and remembers the true beauty he saw in a previous
existence (254b4-7).That is, he has the experience that was said earlier (249d4-e4) to be the madness and enthusiasm of the lover, who,
seeing beauty here, is reminded of beauty there. As a result of this
vision, the charioteer experiences reverence and fear (254b7-8).
Approach is followed by retreat. The charioteer is now compelled to
pull back strongly on the reins so that both horses sit back on their
haunches. The white horse obeys willingly and without resisting; the
black horse obeys but much against his will. The two horses then
retreat (254b8-c4).
The two horses react differently after the retreat. The white horse
experiences shame and terror, and he waters the whole soul with
sweat (254c4-5). This horse experiences not aidos, the good kind of
shame that restrains him from leaping upon the beloved, but aischyne,
shame at having done wrong in yielding to the black horse. When
the black horse recovers from pain, he becomes angry and abusive
and tries, without success, to force the others to approach the beloved
again. Finally, the black horse grudgingly agrees with the others to
postpone a second approach (254c5-d2). This stage of the conflict
ends in a temporary truce.
The whole process of approach and retreat is repeated a second time
(254d2-e5) and many times (pollakis, 254e6) thereafter. The black
horse again compels the others to approach, pulling shamelessly to?
ward the beloved, and the charioteer again pulls back on the reins.
On these subsequent occasions, however, the charioteer's experience
is more powerful (eti mallon, 254e1), and he pulls more strongly on
the reins of the hubristic horse (eti mallon, 254e2). The white horse
is not mentioned.
At last the black horse is tamed (tapeinotheis). He ceases from hybris
and obeys the charioteer, feeling extreme fear at the sight of the
beloved (254e6-8).
The final result of the whole process is that the whole soul of the
lover follows the beloved with aidos and fear (...254e8-255al). This process results not only in the taming of the black horse but also in a permanent - or at least semi-permanent - agreement of all three parts of the soul. The charioteer is in command,
and the two horses obey willingly. The whole soul now experiences
both the aidos that was at first a characteristic of the white horse
alone and the fear (dediduian, 254e9) that the charioteer originally
experienced when he remembered true beauty at the sight of the
beloved (edeise, 254b7) and that the black horse experienced while
being tamed (254e8). The motion toward the beloved that the black
horse once forced upon the others has now been imparted to the
whole soul so that it follows the beloved in orderly fashion. In this
way, the lover, having transformed disorder into order in his soul,
begins to follow his god.
(Belfiore: 192-4.)
