Explaining inconsistencies in literature in thesis and/or supporting information of journal article
With regard to your thesis, you should be as explicit and verbose as possible. It's your thesis, and you will likely never have another chance to explain this issue as clearly as you will now. Avoid being accusatory, but show all the various interpretations of the model in the literature that are relevant and explain your insights thoroughly. Use as many pages as you need, it's not like they charge by the page :)

For the journal article, I would describe as clearly as possible your interpretation of the model and the application to your data, beginning with the best reference available, and adding or clarifying only what is necessary for the results you are describing in the article. There is no need to explore all of the other interpretations. If a reviewer raises concerns, deal with it through the editorial reply process. You could send a concerned reviewer a draft of that section of your thesis, if necessary.

FWIW, I was in a similar position and followed this plan, and I did not get pushback from journal reviewers on this point.