These sorts of adapters, those without an external power supply, are typically designed only for 2.5" drives and smaller. The smaller drives require much less power, and typically run off a single 5V supply.
The reason that the adapter you picture doesn't have a supply is that it has a 44pin mini-IDE connector which includes +5V/GND on the extra 4 pins (on top of normal 40pin IDE). A 2.5" drive would be happily powered from such a converter because it wouldn't require the extra rails you question about (*).
The power requirements are much higher for 3.5" drives than these and similar adapters are designed for. Additionally the 3.5" drives will require 12V for the much large motors that are used, and these adapters will not produce that voltage rail. This is why your USB adapter can't spin up the drive (nor would the IDE/SATA pictured).
As you are interested in only 2.5" drives, such adapters will be workable, they will provide the necessary voltage rails.
For 3.5" drives (and any drive in fact) I would suggest getting a module to add an eSATAp connector to your computer. You can get drive bay modules and PCIe modules that allow you to connect an external drive via this port. The nice thing about the eSATAp variety is that it also provides 12V and 5V power rails, so you can get an passive cable to convert eSATAp to SATA+Power. It also doubles as a USB port when not needed for SATA.
(*) Having said that the adapter appears to have a linear regulator for regulating a 3.3V rail. But this still wouldn't get a 12V rail so wouldn't help connect a 3.5" HDD for those who need such a use case.
Best Answer
These were early hybrid desktop drives when people did not have sata power cables inside their desktops, so they gave you a choice of power connections, Do Not connect both of them at the same time! It will smoke the drive, don't ask how I know this... WD gave no clear warning on the consequences of doing it either.
The Molex connector was discontinued on Sata drives for obvious reasons. They started shipping sata to molex adapters with sata drives instead.
FlexPower: Connector technology that accepts power from either industry standard or new SATA power supplies.
See this PDF