There is an article by Rav Yair Hoffman on the main site about Parshas Haman. He discusses it in general, and then he writes about the minhag of saying it today. I have copied and pasted that part. TLIK, I know you wrote that you don’t consider “I saw it online” as a valid source, BUT he does bring sources that anyone who wants to can look up. And he also explains why it is not possible to find the main source.
THE TUESDAY OF BESHALACH MINHAG
A CAVEAT FOR EVERYONE
– See more at: http://www.theyeshivaworld.com/news/headlines-breaking-stories/522899/parshas-hamahn-and-hishtadlus.html#sthash.4X21NCgf.dpuf