How can one differentiate between mani (spermatic fluid), wady (thick white discharge), and madhy (pre-seminal fluid) if there is wetness on the clothes after waking up for Fajr prayer, especially when there is no memory of a wet dream or lustful thoughts, and what are the signs of mani that necessitate ghusl (ritual bathing)?

Semen is recognized by distinct characteristics: it is thick and white, and its odor resembles the spathe of a date palm or dough. If you are unsure about what you see on your clothes or body, the prevailing fatwa (definition) is that you choose and assign it one ruling from among what you are unsure about. What you mentioned is not among the characteristics of semen.