It seems trivial, and indeed it is, to specify a different configuration file for the mysql client, from the standard .my.cnf to a different file. This is generally needed when you don't want the password to be visible in the process list, but at the same time you don't want to use the standard .my.cnf file.
Despite trivial, I was not able to understand which option was the correct one, and for some strange reason Google didn't help, so this post is mainly to fix this issue.
So basically you can specify a different file with the option
--defaults-file=filename. You can alternatively add the file to the list with the option
--defaults-extra-file=filename. In the latter case, the file will not take precedence on already existent my.cnf files.