Yes the 40-pin PATA is for normal 3,5" desktop-class harddrives that use Parallel ATA (PATA).
The 44-pin PATA is for notebook (2,5") small harddrives, and have 4 extra pins and the pins are closer to eachother. So this is not compatible with 40-pin PATA.
You can buy a real SATA->44-pin PATA adapter. But another trick would be to buy a PATA cable that has 40 pins on one side, and 44-pins on the other side. You use this cable on your current SATA->40-pin PATA adaptor and it should work. Picture of such a cable:
PATA 44-pin (left) to PATA 40-pin (right) cable.