The United States “took ownership” of the Vietnam War under President Johnson through a process that has come to be known as “mission creep.” What this means is that the US gradually took more and more ownership of that war.
At the time when the US Congress passed the Gulf of Tonkin Resolution, the US did not have a large military presence in South Vietnam. Within the next year, that changed. First, the US started to increase the air power that it had in South Vietnam. It then began a sustained campaign of bombing (called “Rolling Thunder”) against North Vietnam. After that, the mission expanded a little. First, American ground forces were brought in to protect American air bases and other American personnel. Next, the American ground forces started to expand their concept of protecting bases. Instead of passively defending American bases, they started to go out on offensive operations.
In short, there was not a single event in which the US took ownership of the war. Instead, it gradually took more and more responsibility until eventually it was fully committed and in complete ownership of the war.