Cilantro is the herb, coriander (seed) is the spice/dried powder. Often you can tell by what you are making and how it’s being used/added, but typically they are differentiated as above in American recipes.
Genuinely confused as well about the pepper, a bell pepper is a pretty universal name for it as far as I knew. Folks also refer to them as green/yellow/red peppers here, or sweet peppers occasionally (usually when used in Italian food), but bell pepper is the generic name.
Arugula is known as Rocket in most of the rest of the world.