There's quite a bit of evidence to support that at least part of the climate system is chaotic.
According to the IPCC:
The climate system is particularly challenging since it is known that
components in the system are inherently chaotic; there are feedbacks
that could potentially switch sign, and there are central processes
that affect the system in a complicated, non-linear manner. These
complex, chaotic, non-linear dynamics are an inherent aspect of the
There's also evidence to support this from an Iowa State University course:
The climate system is thought to possibly have such multiple states;
that is, the climate (or some components of the climate system) could
be stable for a period of time and then abruptly, and for no overtly
evident reason, change to another stable regime. There is evidence
that certain components of the climate system have done this. The
circulation in the north Atlantic Ocean is believed to have gone
through an abrupt change in which the Gulf Stream, instead of tracking
northeastward off the East Coast of the United States and heading
toward Scandinavia, at one time switched very abruptly over about
fifty years (that's abrupt on geological time scales) to an easterly
direction toward the Mediterranean Sea. This caused an abrupt cooling
of the climate in Scandinavia.
In summary, almost-intransitivity is an inherent characteristic of the
dynamics of the climate system that may, for seemingly small or
unknown reasons, launch some component of the climate system into a
pattern not previously seen.
Even more information is given in this NASA post.
To sum all of this up: yes, at least some parts of the climate system are chaotic. But is all of it chaotic? I'm not sure, but I really don't think so.
But as to the definition of a chaotic system, yes, climate is chaotic. There are definitely some things in the climate system that when changed minutely can cause everything to shift. We don't understand everything about climate right now, so it may not be truly chaotic. But as of our research right now, it is chaotic.