The microcontroller is connected the two phone lines through a transformer as show in Fig. 6 in the first pdf file you posted.
Either phone line can be opened by the relay to prevent the call from ringing the telephone. The TIP and RING lines are interchangeable as far as the telephone is concerned.
EDIT: You do understand that the PIC circuit in your article reads DTMF signals, not the caller ID signal which you are trying to decode? The caller ID does not use DTMF, it uses a digital FSK signal consisting of 1200HZ (Logical 1) and 2400HZ (Logical 0) frequencies. Google "caller ID circuit" to find many such circuits.