Besides the named right granted to the operator (through the group), please may you check the workflow folder security tab (properties), in case of for this folder the checkbox would be set to read-only privilege.
But yes, Web client should be behave exactly the same as Windows client, you are right. But I have often noticed differences of object management between web client and windows client...One day, Adobe R&D should do better object management on web version, but due to ACS product, I doubt that it would be done, probably ACS is becoming more or less the web client....
Can the users modify a workflow ?
I think that the write data should be given, for also executing/stopping etc to the given operators group.
Otherwise, I can't help more, sorry, I don't see what should be checked elsewhere.
No Approver can't Modify the workflow and also can't execute/Stop the workflow.So can't provide Write access here
Your approver has the workflow named right. This allows user to execute workflows. Maybe in your windows version he did not have access on the folder.