Convictions are truths that you believe to the utmost that you are able. You would die rather than deny these beliefs, and they shape how you view everything else in the world. If forced to give up a Conviction, the fundamentals of your life are shaken or destroyed.(Ex: "I know that my Redeemer lives.")
Beliefs are beliefs that you have confidence in, but if they're proven wrong, you'll alter them. You may choose not to die for your Beliefs, though you might make lesser sacrifices to support them. If forced to give up a Belief, you are shaken at a deep level, but can generally continue on with life as it was. (Ex: "God means for me to be married to this man.")
Preferences are beliefs that are influenced by how the world makes you feel. They affect your day-to-day life, and may change without much fanfare. You may or may not make sacrifices to accommodate your Preferences. If forced to give up a Preference, you might feel irritated for a period of time. (Ex: "Hymns are better for worship than praise music.")
I am Convicted that my Convictions make me part of Christ's Church, an eternal spiritual body of people assured of salvation after they die. Therefore, I share membership in this Church with people who I thoroughly disagree with at lower levels of belief. We are all working - or being used by God - for the same end, which is God's glory. I owe my brothers and sisters in Conviction all of my love and support.
My Beliefs lead me to membership in the Baptist denomination, rather than Catholic, Methodist, etc. It leads me (reluctantly) to be a Calvinist. I think that it can be difficult to work effectively with people I do not share Beliefs with - more accurately, I think other Church members would be more effective if they shared my Beliefs. ^_^ I would be uncomfortable supporting the teaching of Beliefs that I think are wrong, which is why we struggled with the idea of joining the Presbyterian church. Things that are Preferences for us are Beliefs for them, and some of our Beliefs differ. But if we were in an isolated place with only one church, I would gladly join that church as long as they supported my Convictions, even if they didn't support my Beliefs.
My Preferences lead me to membership in a specific congregation. I like the way a church worships, enjoy a pastor's more intellectual sermons, like the classes offered, like the social activities provided. I think other Church members would be more effective if they shared my Preferences, but I'd never admit that out loud, because it'd be silly. I'd pretty readily join a church that did not conform to any of my Preferences if it supported my Beliefs and Convictions, though I might still look for a church that met my Preferences as well.
Sometimes my Convictions are only others' Beliefs, or even Preferences. (I Believe/Prefer that how mine are sorted is how they should be sorted, depending on the particular belief.) I think that church denominations that consider other churches to be "out of communion" with them are making statements about how their Conviction set doesn't match the other church's Conviction set. This will become more common among denominations, like Catholics, who merge a lot of what I consider "Beliefs" with their Conviction set. (The more vital, fundamental beliefs you have, the fewer people will match that list.)
But like I said, most of what makes me Baptist is on my Belief list, and not my Conviction list. To continue the Catholic example - picking on them just because I'm more informed about them than about Anglicans or Lutherans - I hold a Belief that when the Bible refers to the authority of the Church, it refers to, variously, the authority of the local congregation, or the wider Conviction-sharing Church, and not a single denominational authority structure. I believe that Catholics hold a Conviction that their denomination is the authoritative church. For me, that's not a Conviction mismatch, so I feel "in communion" with Catholics. But for them, I deny one of their Convictions, and so am out of communion.
For another minor example, transubstantiation and the variations thereof are a Preference for me. I prefer the belief that the significance of communion is in the action and the spiritual state a person brings into it, and that no physical or spiritual transformation occurs to the bread and wine/juice. But this belief isn't one I believe in strongly enough to even merit Belief status. If my church has a good argument (that jives with our shared higher beliefs) for transubstantiation, then I'd probably go along with it. I believe that this is a Belief or even a Conviction for some other denominations.
I think that the CBP system makes for a good way to evaluate the fit of a church for me.