Descartes talks about clear and distinct perceptions, in which clear means 'what is present and accessible to the attentive mind' and distinct means 'being clear and sharply seperated from all other perceptions so that it contains within itself only what is clear'. The clear part I can understand, but I have difficulties with really getting the distinct part. In the Principles of Philosophy Descartes does mentions an example about pain that shows that a perception can be clear without being distinct, but this doesn't really help me either.
I assume the (implicit) question being asked here is what "distinct" means in this context, in a more accessible fashion than the quote presented in the question.
Note that from the definitions quoted, the set of perceptions that are distinct is a strict subset of the set of perceptions that are clear. Hence, a distinct perception must be clear, in addition to one's having the ability to distinguish between different distinct perceptions.
As for the second part of the definition of "distinct", this introduces the requirement that when one removes all other perceptions from consideration, the one perception that remains must still be clear, and must not contain anything else which is not clear.
Suppose each element of Set A and Set B is clear or definable and does not have any common element they are distinct. Since ∅ is a common subset of A and B, in this case, while defining the term 'distinction' we can't use the term common subset.
Strictly speaking, since set of natural numbers includes in the set of rational numbers, we can't say set of integers is distinct from set of rational numbers.
I believe this might be the implication.