Al Kafi has two views on this subject.
1) He claimed Imamate
2) He did not claim Imamate.
Then Shaykh Saduq and Shaykh Mufid took the second view and made the prominent view of the 12er Shia. In addition Shaykh Mufid borrowed from narration from Maqatil At Talibeen and that become the primary source for Seerah on Imam Zayd for the 12er Shia.
Here is another hadith on Imam Zayd claiming Imamate.
H 927, Ch. 81, h 16
Muhammad ibn Yahya has narrated from Ahmad ibn Muhammad from
َث al-Husayn ibn Sa‘id from al-Husayn ibn al-Jarud from Musa ibn Bakr ibn Dab
from the one who narrated to him from abu Ja'far, recipient of divine supreme
covenant, who has said the following:
“Zayd, ibn Ali ibn al-Husayn, once went to see abu Ja'far,
Muhammad ibn Ali, recipient of divine supreme covenant. He
carried with him letters from the people of Kufa who invited him
to their (city) and informed him of their unity and collective
decision to ask him to rise. Abu Ja'far, recipient of divine
supreme covenant, then asked him, ‘Are these letters their
initiative or a response to what you had written to them and
asked them to do?’ He said that it was their initiative; they
acknowledge our rights and because of our being from the
family of the Messenger of Allah. It is also because of what they
read in the book of Allah, the Most Majestic, the Most Gracious,
about the obligation to love and obey us. Also it is because of
the suffering impediments and afflictions that we had been
through.
“Abu Ja'far, recipient of divine supreme covenant, said to him,
‘Obedience is an obligation from Allah, the Most Majestic, the
Most Gracious. It is a tradition, Sunnah, that He had established
in the people of the past and in the same way it will continue in
the later generations of the people. Obedience is only for one of
us and to love is for all of us. The command of Allah applies to
His friends because of the rules already made available and
because of the decision already made distinct, because of the
finalized decision and of the measurement that has already taken
place and because of the appointed time on a certain date. Do
not be hasty; Allah does not become hasty due to hastiness in the
people. Do not try to surpass Allah; in your doing so, misfortune
will defeat and destroy you.
“The narrator has said that Zayd became angry and said, ‘The
Imam from us is not the one who sits in his home; draw the
curtain around him and hold back from struggle (jihad). The
Imam from us is the one who safeguards his dominion, fights for
the cause of Allah a real fight, defends those who hold high
regard for him and his rights.’
“Abu Ja'far, recipient of divine supreme covenant, then said,
‘O brother, do you really find any of the things that you
mentioned about yourself? If so, then can you show supporting
proof from the book of Allah or evidence from the Sunnah of the
Messenger of Allah or give a similar example? Allah, the Most
Majestic, the Most Gracious, made things lawful and unlawful.
He has sanctioned the obligations, provided examples and set up
traditions Sunan. He has not made the Imam, who rises with
Divine Authority on His command, engulfed in doubts in the
matters of the obligation of obedience to him. He has not made
him as such that he would act upon an issue before its proper
place and struggle for something before it is there yet. Allah, the
Most Majestic, the Most Gracious, has said, ‘Believers, do not
hunt when you are in the Holy precinct. . . .’ (5:95) Is hunting an
animal that has been made prohibited during Hajj more serious
or the killing of a person that Allah has prohibited? Allah, the
Most Majestic, the Most Gracious, has assigned a place for
everything. ‘Once the restrictions of Hajj are over, you may
hunt. . . .’ (5:2) Allah, the Most Majestic, the Most Gracious, has
said, ‘Believers, do not disrespect the reminders of Allah, the
sacred months, . . .’ (5:2) He has made the number of months
well-known of which four are sacred ones. He has said, ‘. . .
during the four sacred months, they (pagans) may travel
peacefully through the land. You (pagans) must know that you
cannot make Allah helpless . . .’ (9:2) Then Allah, the Most
Holy, the Most High, has said, ‘When the sacred months are
over, slay the pagans wherever you find them. . . .’ (9:5) He has
assigned a place for it. He has also said, ‘Do not decide for a
marriage before the appointed time is over. . . .’ (2:235) He has
assigned a time for everything and for every period of time there
is Kitab (a rule). If you may have the necessary evidence from
your Lord, certainty from yourself in your affairs and you know
well about it then you may act accordingly. Otherwise, do not
aim at an issue in which you have doubt and uncertainty. Do not
act to end a kingdom the sustenance of which is not yet
exhausted; its time has not ended and its Kitab (rule) has not
reached its time. When its time comes, its sustenance diminishes
and its rule reaches its time, the gap will be removed and the
system will follow and Allah will make the followers (of
falsehood) and those followed to suffer humiliation.’
“O brother, do you want to revive the ways of a people who
have rejected the signs of Allah, disobeyed His Messenger,
followed their desires, without guidance from Allah, claimed the
Khilafa (Leadership with Divine Authority) without authority
and evidence from Allah, or a covenant from the Messenger of
Allah? I pray to Allah to grant you, O brother, refuge from being
crucified tomorrow in al-Kunasa.’ Then his eyes became flooded
with tears that flowed down and then he said, ‘Allah is between
us and the people who disregarded our honor, ignored our rights,
made our secrets public and have ascribed us to someone other
than our grandfather. They have said about us what we ourselves
do not say.’”