Do I have to apply for the Luxembourg student permit from my home country?

The student route starts with a temporary authorisation to stay that you request from abroad, on plain paper, to the Immigration Directorate or a Luxembourg diplomatic mission. The official page is explicit that applications lodged from within Luxembourg are inadmissible, so the sequencing has to be respected.

Processing normally takes around 60 days, and if your file is incomplete the clock pauses until the missing items arrive. Visa-required nationals then obtain a type D visa using the authorisation, travel to Luxembourg, declare arrival at the commune within three days, complete the medical and TB check, and submit the residence permit application within three months.