In an effort to respond to your question I found some interesting results. For the object/content you're trying to add the visibility condition for, click on the "DISPLAY" tab and set the condition in the "Set Visibility" section.
In terms of the logic for the Director visibility set the formula to IF($[DC_Manager] = "APPD", TRUE, FALSE) oddly enough after saving the change a red warning displays stating "Formula references a deleted cell or column" however when I run the app the visibility condition is working properly. When the $DC_Manager is set to APPD the director field displays and if I select a different value the director field is hidden.
I decided to test the same formula but instead of using a variable for the DC_Manager I used the actual field data and updated the function to IF([DC_Manager] = "APPD", TRUE, FALSE) and the error warning is no longer present and the Blue formula indicator is present indicating the formula is correct.
I hope this helps!