My hard drive doesn't gets recognized even though it works.

It is a cable problem, these fail very easily in this model. Try a new cable, it should work then.

When you partitioned that new drive in the other System what OS was it? was it a Mac as well? Windows based machines will format depending on the size of the disk either NTFS (Natively non-writable for Macs) or FAT based (Fat/Fat32) Smaller drives and EXFAT for much larger drives. Fat based partitions are read/writable between Windows and Macs however Cannot maintain the OS for either system.

When using MacOS you should use the Disk Utility provided in either the Recovery Method or the Native OS to partition the drive to the Mac Formats (Journaled ..... ) to put a MacOS on it.

Another thing to mention if you can use the same cable on another OS and the drive works/partitions fine, it is highly possible the port you are connecting to is bad (USB) if you are trying internal cables and they arent working, you could have disconnected the cable from the Logic board it could have been damaged over time..

Try to single down where the issue is, using same cable on both machines does 1 work and the other doesnt? if i change ports on the machine it doesnt work on did it start working? etc... Once you can narrow down the issue it gets much easier..

1 More thing to mention some SSDs have firmwares associated to them, i would make sure that it has the latest available firmware attached to it as there have been known instances where the SSDs do not work well in macs without the firmware update.