Given that you have a "Tested" state, it seems that the transitions have been customized. Ensure that all transitions to the "Closed" state have the correct rule on them to update the "Closed By" field. The Closed By definition on the transition should look something like this:
<STATE value="Closed">
<FIELDS>
....
<FIELD refname="Microsoft.VSTS.Common.ClosedBy">
<ALLOWEXISTINGVALUE />
<COPY from="currentuser" />
<VALIDUSER />
<REQUIRED />
</FIELD>
....
</FIELDS>
</STATE>
More information on customizing work item transitions can be found here and here.