Does performing the Fajr prayer immediately upon waking up, and after its prescribed time has ended, count as performing it on time or as making it up?

Whoever performs the prayer outside its designated time is considered to be performing it as a qada' (making up a missed prayer), not as an ada' (performing it on time). If the delay is due to a valid excuse, there is no sin. Otherwise, the person has committed a grave sin, for which Allah has warned, "But there came after them a succession of evil generations, who gave up prayers [or made them of no account] and followed lusts. So they will soon encounter Ghayy (a valley in Hell or ruin). Except those who repent and believe and work righteousness." Among the valid excuses is unintentional sleep, such as one who slept after instructing someone to wake them but they failed to do so, or their alarm clock malfunctioned; in such cases, there is no sin upon them.