I think that everyone gets these two meanings mixed up and mixed together a lot. In spite of this, it generally seems that most people tend to favor convention, and a much smaller group of people tend to favor conviction. Like Tevye, I think they’re both right. Or at least, both are necessary.
Sadly, I think people in both camps tend to demonize much more than they accept or appreciate the other camp. Those who care for convention say of those with conviction that they are (to borrow a meme from fark.com – see this thread for examples) “trying to be special”. Those with conviction call those who care for convention “sheeple” (for examples, see the Wikipedia entry for “Goth Subculture”).

I can understand why the conflict is present and I even suspect there are ways in which it should be seen as healthy. What’s more, I think “doing the right thing” is more than a question of “what”; it’s at least also a question of “when”. So there’s more to the story to be investigated, but it seems incredibly useful to remember that what may appear to be “doing the wrong thing” may very well be someone’s very genuine attempt to “do the right thing”.