Let's review a few Scenarios
Scenario 1
In Acts 8, Peter and John went to Samaria, baptised those people who had accepted the word of God, it reads:
15 When they arrived, they prayed for the new believers there that they might receive the Holy Spirit,
16 because the Holy Spirit had not yet come on any of them; they had simply been baptized in the name of the Lord Jesus.
17 Then Peter and John placed their hands on them, and they received the Holy Spirit.
The baptism didn't mention water, but the Holy Spirit.
Scenario 2
In his speech Peter defended the baptism of the Gentiles, the family of Cornelius, Acts 11:15-16 read:
15 “As I began to speak, the Holy Spirit came on them as he had come on us at the beginning.
16 Then I remembered what the Lord had said: ‘John baptized with water, but you will be baptized with the Holy Spirit.’
The baptism didn't mention water, but the Holy Spirit.
Scenario 3
In Acts 19, when Paul was in Ephesus, he had a dialogue with some disciples, it reads;
2 and asked them, “Did you receive the Holy Spirit when you believed?”
They answered, “No, we have not even heard that there is a Holy Spirit.”
3 So Paul asked, “Then what baptism did you receive?”. “John’s baptism,” they replied.
4 Paul said, “John’s baptism was a baptism of repentance. He told the people to believe in the one coming after him, that is, in Jesus.”
5 On hearing this, they were baptized in the name of the Lord Jesus.
6 When Paul placed his hands on them, the Holy Spirit came on them, and they spoke in tongues and prophesied.
The baptism didn't mention water, but the Holy Spirit.
Review
Now, "What kind of baptism was administered in Acts 16:33?"
It was the baptism with the Holy Spirit. Whether water involved was secondary. But if water involved was no surprise, as the jailer's did get water, when we read
33 At that hour of the night the jailer took them and washed their wounds; then immediately he and all his household were baptized.
Immersion in water was unlikely, as it was said the jailer's household got baptized before Paul and Silas was brought into the jailer's house.
34 The jailer brought them into his house and set a meal before them; he was filled with joy because he had come to believe in God — he and his whole household.