I think the most helpful way of thinking about it is that being baptized into the name of Jesus includes being baptized into the name of the Holy Spirit and the Father. After all, Jesus was the one who baptized with the Holy Spirit (Mark 1:8), and Jesus says that he and the Father are one (John 10:30). I don’t think that you can separate them - to be baptized into the name of Jesus is to be baptized into all three.

What is more important is that people are baptized into the name of Jesus, rather than someone else, like Paul for example (1 Cor 1:13). Jesus is the Christ and we have access to God through him, no-one else.

I think we need to be careful what emphasis we place on baptism, and what it achieves. It isn’t the words that are said at the baptism as much as the attitude of the heart that matters. John 3:16 assures us that we are saved by believing in Jesus, but it doesn’t mention baptism. Baptism is a great thing, but we needn’t be scared that if we say the wrong words we won’t be saved. Jesus’ death and resurrection are stronger than that.

Finally, for what it’s worth, at the church I go to (Anglican), the minister says “I baptise you into the name of the Father, and of the Son and of the Holy Spirit” at our baptism services. They’re clear at that point, but again, it is dependence on Jesus’ death and resurrection for the forgiveness of sin and new life that is what saves us.