You are correct. 3 pin plugs into the motherboard, 4 pin (molex) plugs into the PSU. Your question as to which you should plug your CPU fan into really depends on your fan.
If you are using a 80MM fan or a high RPM 60MM fan (7K+)... you should plug it into your PSU for power. If you plug it into your motherboard, you run the risk of burning out the fan header on your motherboard or the motherboard itself.
The benefit of hooking it into the motherboard is for fan RPM monitoring... many people have their computers set up to give a alarm if the CPU fan dies, that way the CPU won't burn up. Fans these days are pretty reliable and failure is rare.
A good solution is
here (you also need to be sure the fan itself supports RPM monitoring... some don't). It will enable the fan to pull current off the PSU and tie into your motherboard so you can monitor the fan without burning out the header. Be sure you buy a fan with a 3 pin tail. The splitter goes to the PSU for power and hooks into the motherboard for monitoring. If you don't care about RPM monitoring... just get a 4 pin tail.
Case fans plug into the PSU with a 4 pin connector. No need for fan monitoring.