After re-reading the blog a bit and trying what I could with flash, I finally realized that what helps in reducing the file size for the animated gif were two (2) things: First
, reduce the frames per second to one (1). One frame, one second. Second
, reduce the colors in the document (upon saving) to as low as possible (I think I used 16, even). That made my file size just small enough to fit the parameters. There are also tools at http://ezgif.com/
for those who are feeling adventurous, including a .gif maker and a .gif optimizer.
I hope this helps.