How can I get MySQL 5.5 to log warnings to one of the log files?

I have found various things that say that you can log warnings to the MySQL error log, but I have not been able to actually make it happen.

I do have the error log working, and MySQL prints stuff to it on startup and shutdown and occasionally at other times, but if I e.g. SELECT CAST('123' AS DATE); and then SHOW WARNINGS; I can see the warning, but it does not show up in any logs.

I've also tried enabling the general log and the slow query log, but these don't show the warnings either.

I've tried with log_warnings = 1 and log_warnings = 2, but still no warnings are logged.

What am I doing wrong?

mysql> show variables like '%error%';
+--------------------+--------------------------+
| Variable_name      | Value                    |
+--------------------+--------------------------+
| error_count        | 0                        |
| log_error          | /var/log/mysql/mysql.err |
| max_connect_errors | 10                       |
| max_error_count    | 1024                     |
| slave_skip_errors  | OFF                      |
+--------------------+--------------------------+

mysql> show variables like '%warn%';
+---------------+-------+
| Variable_name | Value |
+---------------+-------+
| log_warnings  | 1     |
| sql_warnings  | OFF   |
| warning_count | 0     |
+---------------+-------+
3 rows in set (0.06 sec)

mysql> show variables like '%log%';
+-----------------------------------------+-------------------------------+
| Variable_name                           | Value                         |
+-----------------------------------------+-------------------------------+
...
| general_log                             | ON                            |
| general_log_file                        | /var/log/mysql/general.log    |
...
| log                                     | ON                            |
...
| log_error                               | /var/log/mysql/mysql.err      |
| log_output                              | FILE                          |
| log_queries_not_using_indexes           | ON                            |
...
| log_warnings                            | 1                             |
...
| slow_query_log                          | ON                            |
| slow_query_log_file                     | /var/log/mysql/mysql-slow.log |
...
+-----------------------------------------+-------------------------------+

Edit:

mysql> show global status like 'Aborted%';
+------------------+-------+
| Variable_name    | Value |
+------------------+-------+
| Aborted_clients  | 24    |
| Aborted_connects | 15    |
+------------------+-------+
2 rows in set (0.08 sec)

Edit:

Clarification: I do get [Warning] Aborted connection 1 to db... and [Warning] Access denied for user... messages logged, but not the warnings that you can see via SHOW WARNINGS after e.g. inserting something or running LOAD DATA INFILE... which is what I'm looking for.