Yes, but in the US you have Catholics and all the various sects that fractured from the Catholic church, i.e. Christians who don’t follow the pope. They all follow Christianity, which makes them all Christians, but whether they focus on Jesus Christ and the New Testament or focus on god and the Old Testament is where the distinction lies.
Correct, again its just been my experience that non-catholic Christians will generally refer to themselves as Christians and Catholics will call themselves Catholics over saying they are Christian, at least in the US.