I recently started migrating Workflows from 2010 to 2013. I added the "Workflows I've started" and "My Workflow Tasks" Web Parts to a page. The "Workflows I've started" Web Part works fine but the "My Workflow Tasks" Web Part is just empty although there are existing tasks for me.
In the Workflow settings I renamed the standard task list name from "Workflow Tasks" into "Workflow Tasks Leave Request". Does this make any difference?
When you rename lists, it holds the original list name. So if you code it, you may have problems. I think this is what you are referring to. A good example of this is if you refer to the list in .NET code. You would have to refer to the original name of the list instead of the new name.
I haven't coded anything and I also don't think it's the name of the list. In SharePoint 2010 we had the same configuration and the webpart showed tasks from different task lists. I also added an item to the Standard "Workflows Tasks" list that exists on the site but the task is also not shown in the webpart.
I just tested the workflows with another user and for him the tasks are displayed in the "My Workflow Tasks" webpart. For me the tasks are still not visible although I have higher permissions.
I have delegated the task to the other user and it is directly visible for him in the webpart.