Pre-salvation, my understanding of "slave" was from the American history and from Israel's history in Egypt - both cases involving coercion and brutal dominance, with the slave always watching out for any opportunity to break free!
Post-salvation I learnt of bond-slaves, who turn down freedom, to become voluntary or willing slaves to a master who they've found to be benevolent. I think Abraham's Eliezer might be an example here - he had no thoughts of escape when he was sent on an "errand" with a ten-camel strong, wealth laden caravan (Gen 15:3;24:10).
The difference between my pre and post salvation understanding is, as I've just learnt, a matter of who owns the slave. In the former, the masters used dehumanising methods to coerce the slaves to work and in the latter, the approach was different. Eliezer being the example - an opportunity for escape presented itself and the camels and the wealth they carried would've given him a cushy fresh start if he'd gone on walking! The fact that he went back and restored the remaining jewellery to Abraham's "bank vault" implies that his mind was made up that where he was was where he would die.
The picture painted by Paul's words give that idea, that though in chains, he's happily about his master's mission.
From Pst. Ralph's explanations, I find "saint" to be a synonym of "slave". At salvvsalvation, when we become the temple of Holy Spirit, we become New Creations. Even our nature changes. Where previously we belonged to the rogue angel, with his death-blood, salvation breaks us free from that and in-grafts us onto the true vine (Jhn 15:1). At that very moment, we are set-apart, made "saints". We are then owned by He whose "yoke is easy, and whose burden is light"! Like Peter and company (& Elisha), we leave all, and follow Him. Breaking free would be to return to that other one whose yoke & burden are on steroids!