I have been trying to wrap my head around different definitions of protocol security (stand alone, sequential composition, parallel composition, universal composition) and how the proofs of any of these cases are obtained. However, at current state I find it quite complicated. Can you recommend any good lecture materials or maybe survey articles that would help to get into this topic?
Protocols for Authentication and Key Establishment  By Colin Boyd is considered very good book. That has plethora of examples, proofs etc. in very understandable language and format.
Another book is Modern Cryptography Theory And Practice  by Wenbo Mao. Few chapters have very good introduction to designing protocols in attack-fix cycles.
 Boyd, Colin, and Anish Mathuria. Protocols for authentication and key establishment. Springer, (2003).  Wenbo, Mao. "Modern cryptography: theory and practice." Publisher: Prentice Hall PTR, Copyright: Hewlett Packard (2004).