I've haven't even had a Twitter account for more than a few months, but I've been using www.tinyurl.com for almost a decade, so I guess when I started seeing bit.ly address pop up, I just naturally adapted to using them.
(Not that I've ever actually created one myself.)
It's a bit of hair splitting in the end, I just know what I know from marketing and in our world we don't use Bitly outside of Twitter; user retention is better with branded renditions of the standard "
www.sitename.com" construction when using them in the print world.
Either way, needs to be shorter for the flyers/posters; and Bitly is definitely a strong second choice of solution for that verbose petition URL. You figure people either need to memorize it or write it down. Or worst case they bust out their camera phone (so shorter URL is still easier to get close to and photograph clearly).
We need to treat this as an emotional, point-of-purchase, impulse thing. We get one quick shot to suck the reader in, the path of least resistance is good.