“I have read that if the material is not to be used for profit or redistribution then it may be used. Is using the material on a web site considered distribution or redistribution since it will be placed back onto the Internet in a new context by someone other than the original source? I want to be sure I am in compliance with copyright and intellectual property law.”
Whomever wrote what you read is wrong. Unless an image specifically says that it may be used, and is in the public domain, then you are required by copyright law to obtain permission to use it, and yes, publishing an image on a website is re-distribution (also known as ‘republication’).