How do I make binlog-format=ROW the start up default in the /etc/my.cnf file? I can run the query SET GLOBAL binlog_format = 'ROW' and it works for that session. 

How do I make this setting stick after stopping and restarting MySQL?

NOTE: I am on version  5.5.19 of MySQL (with Moodle) and I just upgraded from MyISAM to InnoDB and therefore need this new setting. 

I have tried adding these two commands (separately) to my.cnf: binlog-format=ROW and binlog_format=ROW and neither work. I have stopped and restarted MySQL every time I changed the my.cnf.

The error I get without this being set is the nefarious error message: 

Debug info: Cannot execute statement: impossible to write to binary log since BINLOG_FORMAT = STATEMENT and at least one table uses a storage engine limited to row-based logging. InnoDB is limited to row-logging when transaction isolation level is READ COMMITTED or READ UNCOMMITTED.
I think your problem may stem from where you wrote the directive

Make sure you place the option under the [mysqld] header in /etc/my.cnf

[mysqld]
binlog_format = ROW


If you place it under any other header, the mysqld process will not see it.

Give it a Try !!!