Looks like some shaley limestone in the bucket. Did you go any deeper, to see if it gets pretty solid..?? More then likely that layer is fractured, and water is perking up through it, off the harder layer below.
I am assuming you are just digging out that mucky spot, and going to replace it with more decent material.
Take it for what it's worth... But if it were me, I'd go up the hill, just about where I thought I saw a pile of spoil on the left side of the picture,and cut the drain across there, and catch the water before it gets to that point in the last picture.
Seeing the terrain in another picture, now..., I'd be tempted to make a complete loop around that spot, and make sure I caught it all. But that's just me talking... Guess I'm thinking, unless you go a good ways to the left looking up the hill, the water may still come in from the side, and still have the same problem.
Again, I'm assuming it is pretty soft either side of your road, and the water is perking out, off that harder seam of rock, across the face of the hill.
Bet that's some pretty nice water though, coming off that limestone aquifer..!!