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Yes exactly. As in plug the bathtub, fill with water, and get in.

Opposite of a shower. LOL

I've seen people call a "bath" just showering but including washing your hair lol.

Here we don't have bathtubs as a norm, people that have them are very few

Yeah. When they say bathing, they are referring to just cleaning, however it is done.

Washing hair? Something I hear about in stories.