What is the difference between User Interaction Design Patterns and User Interface Design Patterns. I have heard that one is the subset of the other, but how do thy correlate?
What is the correct term for ui patterns e.g., from http://uipatternfactory.com/?
I'd say those are User Interface Design Patterns, as they specify how certain UI elements should be designed.
User Interaction as a whole encompasses a little more than just how the UI looks and behaves. In general, User Interaction also relates to the broader picture of how the interaction with the program takes place.
I've heard them used interchangeably, but people might be defining one to be the design patterns used to implement a user interface (for example, using the Command pattern to implement undo and redo operations supported by buttons on the UI and/or keyboard shortcuts) and the other to be patterns that appear in the user interface, such as the Wizard pattern and then defining characteristics about that pattern (name, intent, applicability, known uses, and more).
The site that you linked to appears to fall into the second category - they identify patterns in the interface itself and give them names, show different looks for the implementation of the pattern, and when and why you should use them.
If I had to give specific definitions, I would say that User Interface Design Patterns were patterns defined such as those on the website you linked to, while User Experience Design Patterns relate to the implementation of functionality related to the user experience (for example, my undo/redo example).