'built-in' means that the closet has been designed and built to fit in a particular space within an existing room. It is an antonym of 'free-standing'. There is no indication in the name of whether you can walk-inside it.
A walk-in closet is a small room with a doorway and proper flooring.
In American English, we generally no longer use the word "closet" for closets that aren't "built-in" - set in the wall. We consider those things furniture - armories, wardrobes, etc. "Built-in closet" seems like a redundancy to me. All the things that I refer to as closets are built into the wall.