Hi,
Although I don't work with HP laptops, it is common for the laptops which I have seen, to have some form of connection adapter between the disk drive, and the socket in the laptop. Or to say it another way, the disk drive connection inside the laptop is often not a standard PATA or SATA connector. In these cases, you see a small connection adapter fitted to the drive, and the photo of your drive looks like just such an adapter is fitted.
Therefore, if I was in your situation; if I accepted the various risks of damage which exist (including the problem I explain below); and taking full ESD (anti-static) precautions at all times, I would
gently pull on the two sides of the "non-standard SATA" connector which you can see sticking out of the drive. I expect it will come off in your hand, to reveal a standard SATA connector which will then fit your enclosure. Be careful to pull evenly across the width of that adapter (i.e. do
not pull off one side before the other, as you risk damaging the connector on the disk drive, if you do that). If you are not confident doing this then get help, rather than damage the disk drive, otherwise you will be in a world of pain...

One clue that you have a connection adapter, is that it
seems from your photos that the "non-standard SATA" connector on your drive, is sticking-out significantly from the back of the drive (although a photo from the top of the drive would help to confirm this). A standard SATA connector doesn't stick-out like that. You can search online for photos of a standard SATA connector, so that you can get an idea of where that connection adapter will disengage from the drive.
That's the best I can suggest, without the drive in my hands.

Good luck!