I tried every answer I could find on this issue using MySQL 5.7 on Mac OS 10.12 and ultimately got strict mode turned off not because of the location of my.cnf, which can presumably be in any of the places that MySQL says it checks, but thanks to a UNIX permissions issue.
I used MySQL Workbench 6.2.3.12313 to create my.cnf initially. This caused two possible problems: first, it set the option to "sql-mode" instead of "sql_mode", and it made the file (located in /etc) readable and writable only for root. MySQL does not run as root when you install it the way I did, from the binary package on the MySQL web site--it runs as _mysql. So the _mysql user needs to be able to read /etc/my.cnf, or wherever you put it. In order for that to work, you need to run:
sudo chmod o+r /etc/my.cnf
and for good measure you may also want to run:
sudo chmod g+r /etc/my.cnf
Then make sure to restart MySQL. (I have found that this works best through the System Preferences MySQL panel on Mac OS; using the command line is kind of messy and MySQL Workbench's functionality simply doesn't work.) So long as you have an sql_mode setting in my.cnf that does not involve strict mode, strict mode should be off.