For some packages, in package manager I read that "Canonical provides critical updates for [PACKAGE-NAME] until [DATE]." But what is the formal definition of a "critical update"? For example, if a graphics driver doesn't work properly, would an update solving the issue be a "critical update"? Please cite official documents if possible.
The importance of an update reflects the importance of the bug(s) fixed by that update.
Bug importance is defined at https://wiki.ubuntu.com/Bugs/Importance. A Triaged bug is usually assigned an importance (Wishlist, Low, Medium, High, Critical) based upon the severity of the problem and the estimated number of users affected. Importance helps developers establish the priorities for their work.
Ubuntu generally releases fixes for most High and Critical bugs to the current releases of Ubuntu in the form of a patched package: Foo version 1.1.3 gets patched to 1.1.3a, not to 1.1.4 (that would be a new upstream release).
Graphics drivers (ALL "drivers") are kernel modules, and handled by the Ubuntu Kernel Team. See https://wiki.ubuntu.com/Kernel for their workflow and bug-handling policies.
What is a critical update? critical update(Noun) A broadly released fix for a specific problem that addresses a critical, non-security-related bug in computer software.
Definitions.net
However this can very from manufacturer/vendor/OS as to what they consider critical so it is hard to find an exact definiton based on that, but it seems like your example would fit the definition if the said graphics driver was widely in use...