The "highest 'unofficial' tournament run" that I know is Pete Margo's run of 330 in 1978 or 1979 at the Biltmore Hotel in the World Series of Pool. I say "unofficial" because the run continued after the match was over. However, Weenie Beenie (Bill Staton) was the Tournament Director and Cue Ball Kelly (Carl Zingale) was the referee all through the run. The match was between Pete Margo and Mike Carella. They did not get along well in or out of the pool world. Pete told me that Mike had once made a derogatory remark about him. Anyway, Pete ran 125 or 150 and out to win the match. The crowd asked him to continue the run. Normally, Pete said that he would never do that out of respect for his opponent, but his disdain for Carella prompted him to continue the run. He ran up to 330 and missed an easy 6-ball in the side pocket due to what he said was "physical exhaustion". I have Pete Margo's written documentation and signature to these facts. If there is anyone who has proof of a higher documented tournament run with a referee calling all ball fouls, let's hear about it.
Freddie, as you correctly stated, the actual official highest run in tournament play is 200 by Darren Appleton against Francisco Bustamante at Steinway Billiards during the 2013 World 14.1 Tournament promoted by Charlie Williams. Bustamante made one ball, missed a carom and then Darren ran 200 balls on his first offensive turn at the table. At the end of the match, Darren continued to run another four balls, for a total of 204. Then, with the balls still on the table, Bustamante came over to the table and scattered the balls, so we'll never know if Darren would have or could have broken any other records. Darren said that because he had his next match coming up very soon, he probably would not have continued the run, anyway. I was there and disappointed that the run ended like that: 200 + 4.