In my country we don't call that a bathroom. You can't take a bath in a "half bath". It's just a toilet.
A typical Dutch house has a toilet + sink downstairs for guests or for convenience of the owners. Upstairs there is a bathroom with a toilet, shower, sink and shower for the owners of the home to use.
Because the US traditionally consolidated WC/baths, there was only one room, and people found it more polite to call it the bathroom. Needing to 'use the bathroom' then became a catch all for any function that might happen in there.
But when we started building some without bathing accommodations, the customary terminology didn't go away. Everyone already understood that 'going to the bathroom' really meant using the toilet. So, the term 'bathroom' stuck.
It's called a washroom or a "powder room" in Ontario, Canada. Powder room specifically refers to a small room in a house with only a sink and toilet and is not a term I personally use. A washroom is any room with toilets and sinks, whether or not it has a bath/shower, especially in commercial settings, but "washroom" seems to not be used in the US much or at all, in my experience.
Seems like in (mainland?) Europe people dare to use the actual word a bit better rather than hide the meaning: if the main purpose of the room is the toilet bowl inside of it, it's called a toilet or a wc
A typical Dutch house has a toilet + sink downstairs for guests or for convenience of the owners. Upstairs there is a bathroom with a toilet, shower, sink and shower for the owners of the home to use.