Similarity measure for ordered binary vectors
One thing you could do is fuzzify your vectors: replace each 1 by (for example) 0.4 in its position, 0.2 in the neighbouring positions, and 0.1 in the second position over. Then add up what's in each position. 
With these fuzzified vectors, you can apply a similarity metric either based on a distance or one like cosines similarity. 
Your example would produce: (showing only first decimal)

0000011100 -> 0001378731  

0000011110 -> 0001378873

1000011100 -> 4211378731

cos(x1, x2) = 0.9613,  cos(x1,x3) = 0.9469