Half bath isn’t a mini bath tub. It means just a toilet (and sink) and that the room doesn’t contain a shower or bath tub. Americans don’t seem to like the word toilet and say going to the bathroom (or washroom, or restroom) instead of going to the toilet or loo! So I take it that’s where the half bath phrase comes from??
Mini tubs with a shower over are a thing here though- I think it’s supposed to be to wash kids whilst adults can shower. I don’t get why people don’t just have a proper bath tub though, it’s odd. But you definitely wouldn’t call that a half bath, it’s a full bath meaning you can wash in it!