The IDI is based on the Intercultural Development Continuum (IDC), which was adapted from the Developmental Model of Intercultural Sensitivity. Thus, there are similarities, but they are not the same.
During the development of the IDI, it was found that Integration (or more simply, having a sense of cultural identity) was not a developmental stage indicating increased intercultural competence, but rather characteristics of an identity, of which aspects could be found across multiple stages. Thus, Integration is not a stage included in the IDI.
Similarly, encapsulated marginality is conceptualized as being "stuck" between cultures in some way, and can also be found across various levels of intercultural development.
You can read about the research that took place in the development of the IDC and IDI. There are 2 articles in particular that may be of interest, which discuss these concepts:
The first is Measuring intercultural sensitivity: The intercultural development inventory, which can be found here: https://idiinventory.com/wp-content/uploads/2017/01/IDI-2003-measuring-IJIR.pdf
The second is The Intercultural Development Inventory (IDI): An Approach for Assessing and Building Intercultural Competence which can be found here: https://idiinventory.com/wp-content/uploads/2017/01/IDI-chapter-Moodian-book-2009.pdf (especially, see the footnotes at the end for a discussion about encapsulated marginality)
In addition, there are several other newer articles on our website https://idiinventory.com/ (click on IDI Validation Research in the box on the left side of the page).
Finally, for further information, please review the QA Resources that are available on the IDI Assessment website. They are often updated with new content and so, it should be reviewed periodically.