I'm not quite sure if I'm answering your question as you want, but:
1. From my hazy recollection, the von Mieses "(max) stress" in a structure is the YIELD criterion of the structure; i.e, if the local
stress exceeds that value, plastic flow will occur.
2. Max stress in a structure is simply that: the maximal stress value in the structure.
Clearly, the max stress in the structure must be below the von Mieses yield criterion to ensure that plastic flow does not occur.