PIC Emergency Authority
Could it imply that Wx minimums won't apply?


Yes


  Could it imply that Minimum Altitudes no longer are required?


Yes

Since both of those things are rules under part 91 ("this part") specifically 91.155 
and 91.119 you can indeed deviate from them as per 91.3. 

The answer to "which circumstances" is far to broad to be defined in an all encompassing way which is why the FAA writes it the way they do.