scoring straight (or run) in Cribbage
From the American Cribbage Congress Rules of Cribbage:


  straight (or run), single: Sequence of three or more consecutive cards in any order during the play of the cards; for example, 3-5-6-7-4 (counts three when the 7 is played and counts five when the 4 is played)


When the second 4 is played by Player 1 the last 4 cards played are:

3, 5, 6, 4

This set of 4 consecutive cards clearly meets the definition above of:


  ... Sequence of three or more consecutive cards in any order during the play of the cards ....


Player 1 is entitled to score 4 for this run.

In regards to your claim above that:


  ... he can't count the second run of four cards because they were not in the sequence of play (clockwise). 


Nowhere in the Rules is there a necessity to identify the qualifying cards for a score in a specified order. Your claim otherwise is entirely bogus.

The sequence counts solely by virtue of the required cards, having been played, meeting the criterion of being "Sequence of three or more consecutive cards in any order".  Any such identification of qualifying cards is purely informative, and instructive, to assist beginners in learning the game.