cpp- member type defenition

I am learning cpp. I thought that memeber type are functions/variable which belong to a type. For exanple in the following

class person
{
public:
    string name;
    int age;
};

name and age are member type (in this case member class) of the person class.

Recently I started to read about <chrono> library, and thaen I started to confused. There is for example class system_clock which has time_point member type. But on the other hand there is also class time_point which has Clock member type (e.g class system_clock).

How can a class contains sompe type as a member type and also be a member type of its own member type.

I hope that I succeeded to explain my confusion.