I was able to get Application Pool Identity to work properly, I even was able to get it to access remote SQL Server database successfully. I documented the steps in an answer to another stackoverflow question.
Still, all applications hosted in the same machine are able to access the remote database, which makes sense as I add the SQL Server account with machine name <My Domain>\<My Machine>$
, not account name.
Now, my question is: how to allow only a certain application to access the remote database, not any application on the machine?