I love how the Greek root word for slave is "doulos". I suspect that is where the modern word "doula" comes from. A doula is a hired birth assistant - not a midwife, but the person whose sole job is to cater to the laboring woman's needs. To me, that inspires images of devotion to the work, knowing the importance of it, knowing the uncertainty that can be involved with it, and knowing the level of commitment is complete. A Christian is a "slave" to God in the way that we are compelled to do His work, but joyfully and expectantly.
Paul's greeting is "to all the saints in Christ Jesus"... Saint seems synonymous with "believer" - not just one who believes (because the Bible points out that even Satan "believes"), but one who has accepted and welcomed Christ into his life and is "all-in" in the relationship.
Both "slave" and "saint" describe aspects of our relationship with Jesus. We are devoted and compelled to not just serve Him but delighted to be in relationship with Him.