[jmen24]

One thing to note is that the chimney I mentioned is at least twice the size of the one in your photo. I would look at another price and see what that looks like, especially if you are uncomfortable with the original.

Because the flashing is not visible at the eave were it intersects the chimney I would start looking their. If I had to guess by looking at the photo the water is getting in behind the stucco at the roof intersection with both (roof side) corners of the chimney, running behind the stucco to the shoulder of the chimney and exposing itself then causing the surface water damage below. All that would stem from a flashing issue at the cricket (if it has one).

The damage at the top of the chimney would be caused by an improper seal at the mortar wash and the stucco. The water would seep in behind the stucco and after freezing would break out the stucco in random spots further down the chimney. That is just a guess and would be hard to say for certain without more photos.