I don't think this is the answer you're quite looking for, but posting on the Internet, anywhere written material can be viewed by the public, is, I believe, legally considered publishing, but only from the standpoint of being liable for your words. For instance, if you were to libel or defame someone, then posting on a public forum would be considered publication.
For the purposes of a contest, I doubt posting on a public forum in order to ask for critiques constitutes publication. More likely they have formal publication in mind, whether it's by way of a printed book, ebook, radio program, TV program, etc. The bottom line, however, is that only the contest's administrator could tell you for certain.
Is there any way for you to contact this person or persons to ask? The contest administrators I've known have been pretty nice about replying to questions, and that way you would know one way or the other, rather than simply guessing, or paying an entry fee, only to find out you're disqualified.