The pool mining service just happens to run on the same machine. But the payment address did not come from the client in question.
And besides, I wouldn't expect the pool mining service to cause any "Generated" message in anyone's client. Wouldn't it just come in as a regular payment?
It does show up as "generated". I'm sure that you did use an address from that client for pool mining
Did you use an address from that computer for just a few minutes? That would also explain the low amount.