The reason for the almost universal use of 13V was basically to ensure the devices got switched to programing mode - the spec (back in the old days) called for 12V, but if it was only 11.9V it might not work, far easier to aim for 13V and gain reliability.
Voltage from pin which you use on MAX232 is 8.6V, you use 2 elec-caps and 2 diodes. Vpp generated by that circuit is about 16V - Its too high.
You can use a regulator such as LM317, (resistor: 1K and 220 on ADJ) to generate 13V.