Good news for you - this can all be cleared up simply.
Not Provided is Google saying we are not showing you the keyword data (from organic). There is a filter for a workaround. You'll never know the "exact keywords from organic" that are Not Provided. However, you can filter that data to tell you what pages they're going to. And if you know the page they're going to, it's highly likely that you know the target keyword for that page. So your Not Provided data could actually be those targeted keywords you mentioned.
The filter:
Filter type: custom - Advanced
Field A -> Extract A: Set to Campaign Term: (.not provided.)
Field B -> Extract B: Set to Request URL: (.*)
Output to -> Constructor: Set to Campagin Term: np - $B1
Check field A, field B, and override output field. (leave case sensitive unchecked).
Basically, this filter says "for not provided" - tell me the URL that users are going to. And from there, you can do some digging and determine what keywords are working. Also, this will affect future data, not past data.
I hope this helps.
Cole