Where was the conservatoire entrance exam held?
In chapter 109, Kuroki says, in February, that he is taking care of Rima Namura, as a favour to his old teacher. Rima is in Paris to take a conservatory entrance exam. From details in ch. 110, we can tell that Rima is preparing for the exam in the conservatoire that Nodame attends.

With this in mind, the most reasonable interpretation is that Nodame needed to fly out to Paris to for her entrance exams, and that afterwards (and with the results in hand), she returned to Japan before moving to Paris with Chiaki in March. In her conversation with her friends then, she must have been referring to when she was planning to depart "for good". Her need to go sightseeing when she returns to Paris in March is not too implausible either: perhaps she was on a tight schedule and would not have been able to do any tourist activities while there for the exam.