Yes baptism in water baptizo in Greek which means Immersion. If you have a strong's concordance you can see which Greek word is used there in the original and we have thousands of copies of the originals.

Paul making a mistake about water baptism when he himself was baptized after seeing Jesus on the road to Damascus.

It is evident you haven't read the scriptures lin their entirety for you didn't know about Peter baptizing Cornelius and his family in water or Jesus' great Commission speaking about baptism, or those on the day of pentecost, or Ethiopian Eunuch the Philippian Jailer, or those in Acts 19 or those Paul baptized in Corinthians.

Instead of asking the question: "Water baptism and the indwelling of the Holy Spirit seem to go hand in hand, why the difference on different occasions?" You say "Paul was making mistakes and Acts isn't a doctrinal book book."

Hardly someone I would ever take seriously when it comes to scripture is someone cutting it up to fit a narrative.

And as for Peter saying baptism is not the removal of filth from the flesh he's speaking about the act of going in the water isn't for the physical cleanliness of the person but the spiritual as Paul talks about in Romans 6:1-4. That's why I'm very surprised you used that verse, many people against baptism stay away from it.