The central question
This is whether human 'moral judgment is primarily given rise to by intuition' ? It is not about whether we share morality with (other) animals or how such animals come about whatever moral judgements or moral thinking if any they are capable of.
How Haidt explicates intuition
Haidt refers to the work of Bargh, Damasio, de Waal, and others and tells us that this work stimulated :
me to formulate the social intuitionist model (SIM) of moral
judgment (Haidt, 2001). The SIM posits that moral judgment is
much like aesthetic judgment - a rapid intuitive process - and
defines moral intuitions as follows:
"the sudden appearance in consciousness, or at the fringe of
consciousness, of an evaluative feeling (like-dislike, good-bad)
about the character or actions of a person, without any conscious
awareness of having gone through steps of search, weighing evidence, or inferring a conclusion" (Haidt & Bjorklund, 2008,
The model suggests that moral reasoning is frequent, but
given the speed and ubiquity of moral intuition, moral reasoning
rarely has a chance to play out in an open and unbiased way, as is
often assumed by cognitive-developmental researchers. Rather,
consistent with research on motivated reasoning (Kunda, 1990)
and everyday reasoning (Kuhn, 1991), people engage in moral
reasoning primarily to seek evidence in support of their initial
intuition and also to resolve those rare but difficult cases when
multiple intuitions conflict. (Jonathan Haidt, 'Morality', Perspectives on Psychological Science, Vol. 3, No. 1, From Philosophical Thinking to Psychological Empiricism, Part I (Jan., 2008), 65-72: 69.)
The operative contrast, then, is not with instinct but with reasoning. The OP recognises this but focuses on a contrast between intuition and instinct. But it is impossible to answer the question, 'What is the nature of moral intuition according to Haidt?', when his actual answer involves no reference to instinct whatever.
To probe Haidt's account for its implications for instinct is a perfectly legitimate endeavour. What is not quite right in my view is to ask for Haidt's views about instinct in a question to which instinct is beyond the scope of Haidt's answer. It's no defect in Haidt if he doesn't say anything about the relation of instinct to intuition when he never set out to produce a theory that dealt with instinct at all.
Intuition and reasoning
Haidt does not see an incompatibility between intuition and reasoning. They are not inimical, rather there is a partnership between them. This is clear from the passage above in which it is clear that there are situations where intuitions are not enough and moral reasoning is engaged in. The partnership idea is explicit in the following passage in which he lists three models for the partnership between intuition and reasoning and clearly indicates his preference:
1 Reasoning as senior partner. Intuition and emotion are
acknowledged, but most of the action is in moral reasoning,
which can "channel" moral emotions, and which can and
ought to drive moral behavior. This was Kohlberg's view.
2 Equal partnership. Both processes are (roughly) equally
important in our daily lives, and both can work indepen-
dently to reach different conclusions. This is Narvaez's
view. She described her own efforts to wrestle with choices
about moral action in these words: "Instead of intuition's
dominating the process, intuition danced with conscious
reasoning, taking turns doing the leading" (Narvaez,
2008, p. 235).
3 Intuition as senior partner. Reasoning is acknowledged,
but most of the action is in moral intuition, which can
"motivate" moral reasoning, and which often drives moral
behavior. This is my position, which was shaped strongly
by the work of David Hume, Robert Zajonc, Antonio
Damasio, John Bargh, and Richard Shweder. (Jonathan Haidt, 'Moral Psychology Must Not Be Based on Faith and Hope: Commentary on Narvaez, (2010)' Perspectives on Psychological Science, Vol. 5, No. 2 (MARCH 2010), pp. 182-184 : 183.)
This specifies the nature of intuition, defines the operative contrast as one with 'reasoning' and not 'instinct', and identifies a 'partnership' model in which intuition and reasoning can and do co-operate.
Jonathan Haidt, 'Moral Psychology Must Not Be Based on Faith and Hope: Commentary on Narvaez', (2010)' Perspectives on Psychological Science, Vol. 5, No. 2 (MARCH 2010), pp. 182-184.
Jonathan Haidt, 'Morality', Perspectives on Psychological Science, Vol. 3, No. 1, From Philosophical Thinking to Psychological Empiricism, Part I (Jan., 2008), 65-72.
J. Haidt & F. Bjorklund, 'Social intuitionists answer six questions about morality', W. Sinnott-Armstrong (Ed.), Moral psychology: Vol. 2. The cognitive science of morality, (2008) 181-217.
Jonathan Haidt, 'The emotional dog and its rational tail: A social intuitionist approach to moral judgment', Psychological Review, 108, (2001), 814-334.
D. Narvaez, 'The social intuitionist model: Some counter-
intuitions', W. Sinnott-Armstrong (Ed.), Moral psychology : Vol.2. The cognitive science of morality. (2008), 233-240. Cambridge, MA: MIT Press.
D. Narvaez, 'Moral complexity: The fatal attraction of truthiness and the importance of mature moral functioning', Perspectives on Psychological Science, (2010), 5.
D.Kuhn, The skills of argument. Cambridge, United Kingdom: Cambridge University Press (1991).
Z . Kunda, 'The case for motivated reasoning', PsychologicalBulletin , 1990, 108, 480.