Fortunately the 8400 has a standard ATX12V pinout [See the pinout diagram about halfway down this page:
http://support.dell.com/support/edocs/systems/dim8400/SM/techov.htm#wp1058366 -- and note it matches the standard 24-pin ATX pinout:
http://pinouts.ru/Power/atx_v2_pinout.shtml ], so any standard ATX supply should work. The auxiliary CPU connection does seem to use square sockets rather than the more-typical keyed socket with one side rounded -- but a standard 4-pin CPU power plug should fit (you may need to be a bit "persuasive" -- but not too much).
Note that there's nothing "proprietary" about leaving pin 20 with no connection --> the -5v is no longer required in the current ATX spec, so many power supplies don't supply it. Many earlier Dells did indeed have a proprietary pinout -- unfortunately using the SAME ATX connector, so it was easy to "blow" a power supply or motherboard if you plugged in a standard ATX PSU to one of those motherboards -- but that is not the case with the 8400.
Not sure why Dell would use a non-standard plug on the auxiliary CPU connector; but if you're a bit handy with a jeweler's screwdriver you could easily remove the pins from the 4-pin connector on your new PSU and insert them in the 4-pin connector from the old PSU ==> but look carefully to be sure that's necessary, as it seems unlikely.
A couple other thoughts ...
=> the picture you posted is of the 20+4 pin ATX main power connector -- the 4-pin section of that plugs on to the 20-pin section to make a 24-pin ATX connector ==> that is NOT the 4-pin CPU auxiliary power connector. The 4-pin auxiliary connector will have 2 yellow and 2 black wires leading to it -- is THAT the one you're referring to that doesn't fit on the 8400?? [Here's a pic:
http://www.playtool.com/pages/psuconnectors/connectors.html#atx12v4 ]
=> unless I've forgotten my anatomical details in my old age, the connector on the MOTHERBOARD is the female connector ... and the plug from the PSU is male :-) [r.e. "... the "shapes" of the 4-pin female connector does not match the male connector on the MB."]