The problems you are having with the terminology is because you are connecting two computers together. The RS232 terminology assumes you are connecting a computer or terminal, known as Data Terminal Equipment or DTE for short, with a modem or Data Communications Equipment (DCE).
So the terminal Outputs on (request to send) RTS and the Modem responds with (Clear to send) CTS. As we don't have a modem we have two computers we cross connect the RTS/CTS lines, so from TRS80(25 pin) to PC(9pin) that should be 8=>4, 5=>7.
Then when the modem connects to the other end it raises Data Carrier Detect (DCD) and Data Set Ready (DSR) so we normally connect these to Data Terminal Read (DTR) on the other end. So from the 25 to the 9 pin that would 6+8 => 4 and 20 => 6+1
that's not quite what is shown above,
Code:
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DB9 Ch.A
RX 2 <-------- 2 TX
TX 3 --------> 3 RX
RTS 7 -------> 5 CTS
CTS 8 <------- 4 RTS
DTR 4 ------|--> 6 DSR
|--> 8 DCD
DCD 1 <--|
DSR 6 <--|------ 20 DTR
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This page
http://www.lammertbies.nl/comm/cable/RS-232.html
has lots more info and examples.