Run from iTunes?: A Workflow Service (contains just an AppleScript) created in Automator
Just to cover all bases, I thought I would go through the process you would need to go through in order for this to work. That way you can just do it from scratch and hopefully correct wherever you've gone wrong (or, if it still doesn't work, at least eliminate an error on your behalf as the cause).

In summary you will need to use:


Automator to create a service that runs an Apple Script
System Preferences to assign a keyboard shortcut to the service


Below are the steps involved for each. 

1. Create a service using Automator


Launch Automator (usually found within your Applications folder)
Go to File > New.
Select Service and click Choose.
In the top right hand of the window that appears, ensure that "No input" is selected from the Service receives drop-down list.
Ensure the In drop-down list is showing "Any application".
In the second column containing the long list, find "Run AppleScript" and double-click on it.
This will add the Run AppleScript window on the right.
Now, replace the ( Your script goes here ) with the script you want (taken from the question you reference)
Save the service using a meaningful name (e.g. iTunesFade).


Now to the next step.

2. Creating your shortcut


Go to System Preferences > Keyboard > Shortcuts.
Select Services from the sidebar.
Find your service (it'll be in the list on the right-hand side).
Add a shortcut by double clicking on the service name.
Now go to System Preferences > Security &amp; Privacy > Privacy.
Select Accessibility in the sidebar.
Click on the + sign (you may need to unlock the padlock if it's locked).
Add Automator.
Add Finder (to find this you will need to navigate to /System/Library/CoreServices/Finder.app).
Exit your System Preferences.


Now you should be able to do two things:

1. Run the service from any application by going to the Services list within any Application menu (e.g. Finder > Services, Safari > Services, Mail > Services, etc) and select the service you just created. If iTunes is running, the service will fade iTunes in/out 

2. Use the keyboard shortcut to run the service.

NOTE: I am aware that for some users keyboard shortcuts do not seem to work even though they've been assigned to a service. This is a whole other topic, but if this happens in your case, at least you can run the service from within any application you're using.

Let me know how you go.