Have you tried doing the edge identifications with a physical piece of paper? To make things easier, instead of cutting out a small disk, cut out a large square.
Alternatively, if you want to visualize it using drawings like the ones you've attached, what you could do is cut the square from the midpoint of the left edge to the midpoint of the right edge.This'll give you two rectangles with three edges and part of the fourth edge. The edge identifications on the three edges are aba and aba (resp., and same a and b). Mark down the endpoints of the partial edge as x and y (on both rectangles). Now do the identifications on each piece separately. What you ought to get is two 'incomplete' mobius strips, which when you glue together give you a mobius strip with boundary.