Dear James
I tried as you advised but every thing is remain same; see my following answer please:
1. Make sure alerts work in that affect list. To do this, create a new alert on a test library or on a test list. Perform an action to generate the alert. Check whether you can receive the alert email.
--yes alerts is working in test list and manager receives email
2. Go to SharePoint Designer, edit your workflow, assign the task directly to the manager, save and re-publish the workflow. Check whether the users who created the list item can receive emails after the manager approved the item.
--Yes I have done it:

Creator does not receive any email
3. Make sure the requester is an internal user, as workflow emails will not be sent to external users in SharePoint 2013 workflow by default.
- Yes requester is an internal user
4. Check whether the issue occurs to only one requester or all users who created items in the list.
- No it occurs to all users who created items in the list. No requester receive e-mail
5. Check whether the issue occurs to other lists in this site or other site collections.
- yes it is occurring in all list in my existing site collection.
- then I created another sub site in my site and tested there as well but everything is still unchanged. Manager receives email but requester does not get email after approval or rejection.
Please let me know if you need further information.
Please Please Please help me to sort out this problem, I am stuck here.
Regards
Imran