When I am using Foreign Keys in MySQL, I will get an error if the source value is 0 (because there is no target record with ID 0). Therefore, I am changing the source column to be NULL, and then it works. However, I am not sure if this is the right way this should be done. Is it the right way, or can I somehow keep the source ID set to 0 instead of NULL?
Foreign keys are constraints. This means that if the value of the column that has the foreign key is set to anything (and "anything" does not include NULL), that value must exist in the referenced table or MySQL will throw an error.
So, in short, you can either set the value to NULL, remove the foreign key constraint and set the value to whatever you desire, including 0, or add a record with a 0 in the referenced table. Of these options setting the value to NULL seems the cleanest.