Family matters

My first cousin's wife just gave birth to a beautiful baby girl. Two people - with entirely too much time on their hands - started arguing over the nuances between "once removed cousins" and "second - third- fourth - etc. cousins".

Would that make this bundle of joy my first cousin once removed? My second cousin? Or something else entirely?

And what does that make my dad's first cousin's daughter to me? I always thought she was my second cousin.

Does anyone know?