Thanks for your question!
Matthew 28:19 Go ye therefore, and teach all nations, baptizing them in the name of the Father, and of the Son, and of the Holy Spirit:
In Acts 2 only the name of Jesus Christ is mentioned, but it seems sensible to conclude that one cannot be baptised into Christ without the active participation of the Father and his power.
Acts 2:38 Then Peter said unto them, Repent, and be baptized every one of you in the name of Jesus Christ for the remission of sins, and ye shall receive the gift of the Holy Spirit.
In Acts 8 it is described as baptism in ‘the name of the Lord Jesus’.
Acts 8:16 (For as yet he was fallen upon none of them: only they were baptized in the name of the Lord Jesus.)
In Acts 8 it is described as baptism in ‘the name of the Lord’.
Acts 10:48 And he commanded them to be baptized in the name of the Lord...
In Acts 19 it is described as baptism in ‘the name of the Lord Jesus’.
Acts 19:5 When they heard this, they were baptized in the name of the Lord Jesus.
In Acts 22 it is described as baptism ‘calling on the name of the Lord’.
Acts 22:16 And now why tarriest thou? arise, and be baptized, and wash away thy sins, calling on the name of the Lord.
In Romans 6 baptism is described as ‘into Jesus Christ’.
Romans 6:3 Know ye not, that so many of us as were baptized into Jesus Christ were baptized into his death?
In 1 Corinthians 1 baptism in the name of Christ is implied.
1 Corinthians 1:13 Is Christ divided? was Paul crucified for you? or were ye baptized in the name of Paul?
1 Corinthians 12 we are told we are baptised by, and made to drink into, one Spirit.
1 Corinthians 12:13 For by one Spirit are we all baptized into one body, whether we be Jews or Gentiles, whether we be bond or free; and have been all made to drink into one Spirit.
In 2 Corinthians 5 we are told that God (the Father) was in Christ, indicating that all that is done in the name of Christ is of necessity done in also in the Father’s name. Baptism in Christ involves reconciliation to God.
2 Corinthians 5:19 To wit, that God was in Christ, reconciling the world unto himself, not imputing their trespasses unto them; and hath committed unto us the word of reconciliation.
In Galatians 3 it is described as baptism ‘into Christ’.
Galatians 3:27 For as many of you as have been baptized into Christ have put on Christ.
In Ephesians 1 we are told that everything is gathered together in Christ emphasising the reconciliation between heavenly and earthly things.
Ephesians 1:10 That in the dispensation of the fulness of times he might gather together in one all things in Christ, both which are in heaven, and which are on earth; even in him:
Ephesians 4 brings together The Father, the son (Lord) the Spirit with the one body, one hope, one faith, and one baptism. This seems to me to reconcile the commandment of Christ in Matthew that we started with, with what we have read elsewhere.
Ephesians 4:4 There is one body, and one Spirit, even as ye are called in one hope of your calling;
5 One Lord, one faith, one baptism,
6 One God and Father of all, who is above all, and through all, and in you all.
Everything we do must be done in the name of Christ and thanks given to God by him.
Colossians 3:17 And whatsoever ye do in word or deed, do all in the name of the Lord Jesus, giving thanks to God and the Father by him.
I conclude that ideally we should be baptised in the name of Jesus Christ but with the full formula of the words commanded by Christ being used:
‘in the name of the Father, and of the Son, and of the Holy Spirit’
This is not to say that the form of words is critical in baptism. What makes baptism valid is what we believe when we are baptised. If we understand and believe the truth, repent and are baptised by immersion, then baptism just ‘in the name of Jesus Christ’ will not invalidate our baptism.
There is no ‘magic formula’ in the words used, only a proper understanding of what we are doing matters.
I hope this helps,