I agree with you that here (Luke 12:50) the word "baptism" is not talking about water baptism. I am aware that Jesus was baptized in water earlier, and also in the Holy Spirit. So this could not be water baptism. Here the noun "baptisma" refers to immersiion, and not water baptism.
Jesus is drawing a parallel between water baptism and his death. In water baptism one is immersed for a while and then is raised up. During his baptism of suffering, he would be immersed in suffering and not in water. He would stay immersed for a while and then be raised up from the dead.
However, I believe that the following verses talk about immersion in water:
38Peter said to them, "Repent, and each of you be baptized in the name of Jesus Christ for the forgiveness of your sins ; and you will receive the gift of the Holy Spirit.
Here the Greek verb "baptizo" is clearly referring to water baptism
20who disobeyed long ago when God waited patiently36 in the days of Noah while the ark was being built. In it only a few people, eight in all, were saved through water,
21and this water symbolizes baptism that now saves you also--not the removal of dirt from the body but the pledge of a good conscience41 toward God. It saves you by the resurrection of Jesus Christ
Here the Greek noun "baptisma" refers to immersion. What kind of immersion? v 20 mentiones "water," and v21 also mentions the word "water"
36As they went along the road they came to some water ; and the eunuch said, "Look ! Water ! What prevents me from being baptized ?"
37[And Philip said, "If you believe with all your heart, you may." And he answered and said, "I believe that Jesus Christ is the Son of God."]
38And he ordered the chariot to stop ; and they both went down into the water, Philip as well as the eunuch, and he baptized him.
Here the Greek verb "baptizo" is used twice. What kind of immersion was this? It's clear. Both, v36 and v38 mention the word "water."
47"Surely no one can refuse the water for these to be baptized who have received the Holy Spirit just as we did, can he?"
48And he ordered them to be baptized in the name of Jesus Christ. Then they asked him to stay on for a few days.
Again the Greek verb "baptizo" is used to indicate immersion. But what type of immersion? In v. 47, the word "water" is mentioned.
1It happened that while Apollos was at Corinth, Paul passed through the upper country and came to Ephesus, and found some disciples.
2He said to them, "Did you receive the Holy Spirit when you believed ?" And they said to him, "No, we have not even heard whether there is a Holy Spirit."
3And he said, "Into what then were you baptized ?" And they said, "Into John's baptism."
4Paul said, "John baptized with the baptism of repentance, telling the people to believe in Him who was coming after him, that is, in Jesus."
5When they heard this, they were baptized in the name of the Lord Jesus.
6And when Paul had laid his hands upon them, the Holy Spirit came on them, and they began speaking with tongues and prophesying.
7There were in all about twelve men.
Here Paul came across some men who were water baptized, but they did not even know that there was a Holy Spirit. He later found out that they received John's baptism, which was just a baptism of repentance. He then baptized them in the name of Jesus. How did he baptize them in the name of Jesus? Put two and two together, and you will know that is was with water.
Let me clarify my understanding further: Some believers say that water baptism is John's baptism, and baptism in the name of Jesus is not done by water. If this was true, then Paul would immediately have recognized that since they were baptized with water, it was John's baptism. But since both baptisms, John's and Jesus', were done with water, Paul had to seek clarifications.
Now one might say, "but he baptized them by laying of hands." That's not true! The laying of hands and the Holy Spirit baptism came after the water baptism. They are also mentioned in separate verses.
Yes, he initially enquired if they had received the Holy Spirit. When they answered in the negative, he immediately knew that it was John's water baptism, since the baptism of John did not involve the Holy Spirit and does not save. This means that the baptism in the name of Jesus involves the Holy Spirit, and saves.
By the above verses, we can infer that the other incidences of immersion (Lydia, Jailer, etc) were water immersion as well.
These are my thoughts. As I was studying what you wrote, one verse led to another. Pls let me know what you think.