I work in the area of combinatorics on words and am not aware of any existing terminology for whatthetypeofsubsequencethat you are tentatively calling a "co-subsequence", so you can probably stick with this name if you like.
By the way, Joseph provided a nice list of books on words/strings. Here are two more that I would highly recommend:
M. Lothaire, "Algebraic Combinatorics on Words", vol. 90 of Encyclopedia of Mathematics and its Applications, Cambridge University Press, 2002
J.-P. Allouche & J. Shallit, "Automatic Sequences: Theory, Applications, Generalizations", Cambridge University Press, 2003
The first book and two others in the Lothaire series are freely available here.