How is correct to interface an MCU (Arduino)'s UART with this RS232 TTL device?
The first thing to do is to verify the polarity level of the signals.  The pullups imply the idle level is high, which is normal for logic-level RS-232.  However, the transistor in the first diagram inverts the signal, which implies the opposite.  See what state the transmit lines are in when nothing is being sent.

Also verify that the you are really using logic-level RS-232 on the microcontroller end.  If there is a RS-232 converter chip in there, it won't work right.  You want to connect the microcontroller UART lines to the bill acceptor UART lines directly, assuming the micro follows the usual convention of idle is logic high.

Also, test each direction independently.  Do something with the device that should cause it to send something, and watch its TX line on a scope.  That will tell you polarity, baud rate, and the like.  Then send it some bytes that should cause something visible to happen to test the other direction.