PRNGs which are not CSPRNG
Item (2) is not part of the definition of a standard CSPRNG. If you look at where this definition was taken from in Wikipedia, this item refers to a "forward secure CSPRNG". So, this is a definition of forward security and not the standard notion.

Also, the next-bit test is one way of defining, but certainly not the only way.

In any case, it is easy to construct a CSPRNG that is not forward secure; simply always store the initial state.