Anointed is in reference to what the Holy Spirit does - in Christ's teaching (John 14-16) Jesus teaches his disciples that He is getting ready to leave them but they will not be orphans - why? Because the Counselor/Comforter (Holy Spirit) will come helping them to remember what Christ has taught and allow believers to dwell not only with Jesus but also the Father “If anyone loves me, he will keep my word, and my Father will love him, and we will come to him and make our home with him. Whoever does not love me does not keep my words." (John 14:23-24).
Notice what Jesus teaches when it comes to God showing Himself - "He will keep my word - whoever does not love me does not keep my words." This should be the test for authenticity in the Christian life - do we still hold to what Christ taught or do we move away from it - the Holy Spirit's task according to Jesus is to "convict the world concerning sin, righteousness, and judgment" (John 16:8). Paul later teaches in 1 Cor. 2 that the Holy Spirit teaches "the things freely given us by God" so that we can have the "mind of Christ" which is all rooted in the crucifixion - 1 Cor. 2:2 "For I decided to know nothing among you except Jesus Christ and him crucified." - everything we need is found in what Christ did to set us free from sin at the crucifixion - the Holy Spirit takes what Christ did on the cross and applies that to us now so that we can be new people, born again (regeneration), holy and clean, no longer under condemnation (Romans 8).
Titus 3:4-7 But when the goodness and loving kindness of God our Savior appeared, 5 he saved us, not because of works done by us in righteousness, but according to his own mercy, by the washing of regeneration and renewal of the Holy Spirit, 6 whom he poured out on us richly through Jesus Christ our Savior, 7 so that being justified by his grace we might become heirs according to the hope of eternal life.
Colossians 1:20 and through him to reconcile to himself all things, whether on earth or in heaven, making peace by the blood of his cross. 21 And you, who once were alienated and hostile in mind, doing evil deeds, 22 he has now reconciled in his body of flesh by his death, in order to present you holy and blameless and above reproach before him, 23 if indeed you continue in the faith, stable and steadfast, not shifting from the hope of the gospel that you heard, which has been proclaimed in all creation under heaven, and of which I, Paul, became a minister.
What does it mean to continue in the faith, stable and steadfast, not shifting?
Romans 8:2 For the law of the Spirit of life has set you free in Christ Jesus from the law of sin and death. 3 For God has done what the law, weakened by the flesh, could not do. By sending his own Son in the likeness of sinful flesh and for sin, he condemned sin in the flesh, 4 in order that the righteous requirement of the law might be fulfilled in us, who walk not according to the flesh but according to the Spirit. 5 For those who live according to the flesh set their minds on the things of the flesh, but those who live according to the Spirit set their minds on the things of the Spirit.
Basically if your gospel goes back to a list of do's and don't (according to the flesh - what you do) instead of what God has done you are not keeping in step with the Holy Spirit but "submit again to a yoke of slavery" instead of standing firm in Christ's freedom (Gal. 5:1).
Anointed followers of Christ are people who understand the things freely given that allow them to develop the mind of Christ through the power of His death, burial, and resurrection - it is the Holy Spirit who works the circumcision of our hearts (Isaiah 6:10, Rom. 2:29, Col. 2:11-14).
When a person asks if you think they are a Christian you can go through the teachings of Christ and ask do you follow what Christ taught about who you are and who He is, does your salvation fully rest on Him, or do you use it as a stepping stone to another gospel that moves away from what the Holy Spirit has freely given?