If I run ./exim -d -bt user@domain all seems well, but when I send a message from my User Agent, it does not arrive at its destination.

Try sending a message directly to Exim by typing this:

exim -v user@domain
<some message, could be empty>
.

If the message gets delivered to a remote host, but never arrives at its final destination, then the problem is at the remote host. If, however, the message gets through correctly, then the problem may be between your User Agent and Exim. Try setting Exim's log_selector option to include +arguments, to see with which arguments the UA is calling Exim.