Question 1: does anyone have a satisfactory definition of "grapheme"?
The tightest definition I found via internet searching is on
this page, and is due to "David Crystal":
Graphemes are the smallest units in a writing system capable of causing a contrast in meaning. In the English alphabet, the switch from cat to bat introduces a meaning change; therefore, c and b represent different graphemes. It is usual to transcribe graphemes within angle brackets, to show their special status: <c>, <b>. The main graphemes of English are the twenty-six units that make up the alphabet. Other graphemes include the various marks of punctuation: <.>, <;>, etc., and such special symbols as <@>, <&>, and (£). . . .
Question 2: What if anything is wrong with the following analysis?
I find Mr. Crystal's definition unsatisfactory. Following the definition, one might consider individual Chinese characters to be graphemes in most or all cases. Indeed, we have
here a list of standards for 4700+ common characters published under the title of "List of graphemes..."
For example, the character "帥" (handsome) is capable of causing a contrast in meaning. According to the published list I referenced above, it should be a grapheme.
If we modify that character by adding only one horizontal stroke (一) to the top of the right half of "帥", we get a character "師" (teacher, instructor). Presumably this stroke is a "smaller unit" than a whole character and it is definitely capable of causing a contrast in meaning, so under Mr Crystal's definition it seems that 師 or 帥 is not a "grapheme" at all. Only the extra stroke (一) would seem to meet that definition.
In Chinese, sometimes the smallest such unit is going to be a single stroke. Other times it's going to be a larger structure that still doesn't necessarily constitute a character (generally, 部首). There are also cases in which a full character could meet the definition (e.g., 口). I'm thinking the concept of "grapheme" might just be a sloppy idea resulting from excessive reliance on western languages/thought and/or lack of imagination among the people who invented it. It's also possible that there are superior definitions out there to Mr. Crystal's.