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The serial download cable is described in several of Rev-Ed's pdf files. The Range Summary pdf, the manuals and tutorials for the '08 and the '18... The chart above is based on a summary of that information. ("axe003_manual.pdf", "AXE001_pinout.pdf" & "AXE025.pdf")
If you solder it up yourself, be sure to do a continuity check (with multimeter on ohms, or a beep tester) between all the pins to make sure that none of them are shorting together, even with the cable flexing a bit at the connector, and that each wire connects through from one end to the correct pin at the other end.
One idea is to use the lead of an old serial mouse. Usually they have 4 wires inside, corresponding to pins 2, 3, 5 & 7 of the D9 connector. Identify pins 2, 3 & 5, ignoring the wire to pin 7.
If you can't make up a cable, and don't have an electrically minded friend to make one up for you, you'll be buying the cable from one of the PicAxe distributors near you. Check the list of agents on the PicAxe General page.
Here is a circuit that I use, which allows me to use serin and serout commands to the Serial Terminal window in the Programming Editor. See the Software serial page for examples.
The key difference between my circuit and the Rev-Ed circuit is that I've added two protection diodes, so as not to use the internal diodes in the picaxe ports. For the zener diode I use a 5v1 zener, and for the other diode I use a 1N60 germanium, although any small signal diode will do (eg. 1N4148, 1N914).
The resistor values shown are not critical, but some of their ratios can matter.
Use anything from 4k7 to 22k for the input resistor to ground (shown as 8k2 on my design, and 10k on Rev-Ed's circuit).
Use anything around 10k up to 22k for the series resistor (marked as 10k in my circuit, and 22k in Rev-Ed's circuit).
Use a value around 100k to 470k for the serin pin pulldown resistor (shown as 220k in my circuit).
Note that you don't have to connect the SerIn line to leg_4 and jumper it to SerIn pin like I have, you can just connect it like the the Rev-Ed circuit, just with the extra diode and zener diode for decent input port protection.
Note also that If you wish to connect serin to port-pin 3 (leg 4) then you must provide a protection diode there to prevent overvoltages on that pin, as it is different from the other ports in not having that diode to +V internally. All that's required is to provide any small signal diode (eg. 1N4148) to +V, with the band on the diode at the +V end. The zener in my circuit performs that function, and so the protection diode to +V isn't needed.