Recommended Books
The following books are highly recommended. Some are about game design (and have heavily influenced this theory), others are about related subjects or just good for someone who wants to be a game designer. These books are listed in order of importance for a game design student (although be warned that Characteristics of Games is dense reading and not for a general audience).
Characteristics of Games by Elias, Garfield, and Gutschera (the best hardcore book about game design)
The Art of Game Design by Jesse Schell (the best general audience book about game design)
The Design of Everyday Things by Donald Norman (the best book about general design principles)
Understanding Comics by Scott McCloud (a timeless work about the nature of abstraction)
C.O.D.E. by Charles Petzold (the book to read before you start to learn how to program)
Game Feel by Steve Swink (a foundational work on real-time controls)
Puzzle Craft by Mike Selinker and Thomas Snyder (the best book on puzzle making ever written)
Other Resources
Theories are well-established for non-interactive narrative, art, and music, and can be useful in interactive media as well. There are also engagement theories for students, workers, and more that use the same name. While they are tangentially related to this theory, the focus here on players, viewers, etc. makes the details much different. The main common point is the focus on motivation as being critical to engagement. There are also a lot of game designers who write about game design theory on the internet who are well worth reading:
James Portnow: Extra Credits
Daniel Cook: Lost Garden
Tadhg Kelly: What Games Are
Mike Stout: On Game Design
Keith Burgun: 3 Minute Game Design
Ernest Adams: The Designer's Notebook
Aesthetics vs. Engagement
In 2004, LeBlanc, Hunicke, and Zebek proposed a framework for looking at games called Mechanics, Dynamics, Aesthetics (MDA). What MDA calls aesthetics ("the desirable emotional responses evoked in the player, when she interacts with the game system") is almost identical to the types of engagement used here. The primary difference is that engagement types are not limited to being evoked by the game system (although most designers don't really restrict themselves to this when using MDA), but can be evoked by any element of the work. Another minor difference is that the responses do not have to be emotional--they can be any basic human desire or instinct (although again, most users of MDA would not restrict themselves to this limit either). The other difference is the break-down of the individual types of engagement/aesthetics. Most are the same, but MDA's Narrative is replaced with Catharsis, MDA's Submission is replaced with Accomplishment, Competition is added (James Portnow also adds Competition to his own breakdown in his Aesthetics of Play episode of Extra Credits), along with Flow. What engagement theory really adds in this area are the sub-categories of each type and the particular definitions of each type.
The "elements" of engagement theory could be considered to be in the same category as MDA's "mechanics", although engagement theory takes a much broader view of this category. The "dynamics" category of engagement theory and MDA definitely overlap, but are approached very differently. The composition and other techniques are not part of MDA at all.