I’ll reply with an American view! In the United States I would ask for a restroom or a bathroom. If I am in a public place (like a restaurant or a store) I will usually ask for a restroom. (What a strange word for a toilet!) :) But, if I am with friends, or visiting a person’s house, I will just ask for a bathroom. I think in this case, it is more “polite” to ask for a “restroom.” These are both rooms with a toilet and sink, a place to wash your hands. They may, or may not, also include a tub or shower.
You also ask about the word “washroom.” It is more common to say “washroom” in Canada. This is a room like a bathroom, with a toilet, sink, and possibly a tub or shower. If you said “washroom” in the US, people would understand what you mean, but it is not common.
In my experience “lavatory” is not common. But anyone in the US or Canada would probably understand that you meant “bathroom.”
Savage, you also ask about the word “latrine.” A latrine is a pit dug in the ground that people use as a toilet.
I hope these definitions have been helpful to you. They can be very confusing!