If it is not possible to enter the value how are you able to edit it?
Which layout are you on when the problem happens?
You have the wrong field on your layout. You need to set up a director name field in productions. By placing the field from directors (directors::name) on your productions layout, any changes to the text in this field change the name of a director in your table of directors.
You can't type into such a field until you enter or select a director ID, because you have no link to a director name record until you do that.
BTW, you can use such a name based auto-complete value list to look up director IDs so that your production record is properly linked by ID.