When do you increment the 'Successful run' counters?
Add tokens, then access. There's a timing structure in the FAQ (here). The end of the run goes like this:


  4 The Runner APPROACHES the attacked server.
  
  
    4.1 [Paid abilities]
    
    4.2 The Runner decides whether to continue the run.
        ...Either the Runner JACKS OUT: [end the run unsuccessfully]
        ...Or the Runner continues the run: go to 4.3.
    
    4.3 [Paid abilities and non-ice can be rezzed]
    
    4.4 The run is considered to be successful. (‘When successful’ conditionals meet their trigger conditions)
    
    4.5 Access cards, then [end the run].
  


The run is considered successful (and tokens are added to Medium and the like) in 4.4. You access the cards in 4.5.

To answer the question within the question, this also means you'll get extra cards starting from your second Medium run, and not just starting from your third.