What is the meaning of the rule "attributing an incident to the nearest time" and what are its conditions? Is it obligatory to repeat (the ritual purification) for one who finds a barrier after their ghusl or wudu, but does not know when it occurred? Is it permissible for a woman to choose the type of discharge if she is unsure?

The rule: "The default is to attribute an occurrence to its nearest time" means that if there is no clear evidence for something, it is attributed to the nearest time to it because that is what is certain. So, if a sleeping person sees seminal fluid but does not recall a wet dream, ghusl (ritual bath) becomes obligatory for him, and he must repeat every prayer he performed since his last sleep. An example of this is someone who performs ghusl or wudu (ablution) and then sees something that prevented water from reaching a part of their body. The default is that this occurred after the purification, so they are not obligated to repeat the ghusl or wudu.

As for the discharge that exits from a woman, it is pure but invalidates wudu. Madhy (pre-seminal fluid) is impure. If there is doubt between the two, the Shafii school allows for choice: if the exiting fluid could be semen or something else, one has the choice between them. If one considers it semen, ghusl is performed. If one considers it something else, wudu is performed and whatever it touched is washed.