There are some instances where this will not update when you change the name of the program - and you will need to MANUALLY update these values (otherwise, your smart campaigns will not run properly). This will often occur when the value is contained in a custom field and/or when the field doesn't solely (and only) contain PROGRAM values. For example, when the value is included in these fields, they will always take on the current program name:
- Member of Program ("program" is xyz)
- Change Program Status ("program" is xyz)
Here's an example where the program name always gets updated - for those program values on the left - but not updated for any of the custom field values on the right (and therefore, we always must edit those values on the right after we clone the template to a production program):

The trigger you included in your example doesn't contain any "program" fields (even though they will accept names of program) - and therefore will not update when you make changes to the program name (or clone to another program).