"I assemble the desk."
Translation:Eu monto a mesa.
Sorry, I didn't mean to say you were wrong. Based only on the English translations, I would guess that the difference between them is similar to the difference between these two photos. The writing desk being more likely to be found in the living room at home and a desk found in an office or a home study. The second one looks a bit easier to assemble. Whether this distinction really holds in Portuguese I don't know.
The left picture (the one with the chair and portrait) is definitely a "escrivaninha". It can also be called a "secretária".
The right picture can be called a "mesa". It might be called escrivaninha, but mainly when it's a personal table in a home office. In company offices you would not hear that word.
For "carteira", these are the pictures: