I think that you are confused about what "independent of path" means. For conservative forces the work done between two STATES of the system is independent of path. We also can define a potential energy which is a function of the state of the system. In your case, elastic PE is a function of the state of the spring. This is the system with elastic PE. It is obvious that your (b) and (c) represent two distinct states of the spring so the work done going from (a) to (b) has no reason to be equal to the work done from (a) to (c). The path independence just refers to one pair of states. Not that going from an initial state to any other state you do the same work.
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