How to identify I already own a paid app which is on a 100% discount offer period?
I haven't paid attention to this aspect in previous versions of Play Store but with the current version I see no confusion

If you don't see offer period (as in your first screenshot) you can install or you own the app as you say


Discount offer not availed





Discount offer availed app installed and uninstalled 




The strike through pricing is only shown for the duration of offer. See


  About sale price annotations 
  Earn more revenue with the right monetization options  


Here's an app that I bought on sale many months ago and it seems to have an indefinite offer period and it shows discounted price



Bottom line - ignore the strike through pricing. You are good as long as you don't see the offer duration notice