When I do git diff COMMIT I see the changes between that commit and HEAD (as far as I know), but I would like to see the changes that were made by that single commit.
I haven't found any obvious options on diff / log that will give me that output.
To see the diff for a particular COMMIT hash, where COMMIT is the hash of the commit:
git diff COMMIT~ COMMIT will show you the difference between that COMMIT's ancestor and the COMMIT. See the man pages for git diff for details about the command and gitrevisions about the ~ notation and its friends.
Alternatively, git show COMMIT will do something very similar. (The commit's data, including its diff - but not for merge commits.) See the git show manpage.
(also git diff COMMIT will show you the difference between that COMMIT and the head.)