Italki friends means the people you know from the site Italki, who already developed a friendship with you.
Friends of Italki means the different thing. I mean if you're trying to refer to someone you knew from Italki by this phrase, people can understand what you're meant to say, but strictly speaking, take Jura's sentence as an example:
"held for all the friends of Italki"
That means all of the Italki members here, they don't necessarily have a friendship each other or even know each other in person.