When a culture engulfs you like falling into a large pool of water, you are immersed. Generally, when a person tries to be immersed, they leave behind all tie to another culture.
To become culturally immersed experienced traveler learn to continuously turn away from known experiences, and move toward unknown ones.
The test of being culturally immersed is easy, when you are extremely uncomfortable, very vulnerable, and you have no way of knowing the rules, then you are immersed in a culture. All know cultural norms and standards are going to work, often upon fully being immersed into a culture you will experience the same castigations as other members of the culture.