Is my understanding of the rulings regarding semen correct, concerning the obligation of ghusl if it is discharged with lust while awake, or during sleep with or without a wet dream, or a short while after waking with a wet dream but not otherwise? And is my method of distinguishing between semen and pre-seminal fluid, by means of the smell after drying, correct?

Ghusl (ritual bath) becomes obligatory if semen is discharged while awake with customary desire, or in sleep, whether with desire or without. If a person experiences a wet dream but finds nothing immediately, then semen is discharged hours later, ghusl becomes obligatory according to the majority of scholars. However, if semen is discharged without a wet dream or pleasure, it does not necessitate ghusl according to the majority, differing from the Shafi'i school. Semen can be identified by its smell whether discharged wet or dry.