This is probably not an issue, but I thought I would mention it just for the sake of future knowledge. Many Seagate SATA drives have a small jumper on the back. The jumper on the pins means it is 150 MB (SATA 1). If you remove the jumper, it becomes 300 MB (SATA 2). If the drive worked before the installation of Linux Mint, then the jumpers are not the problem. But you now have a bit of trivia to share at your next dinner party.
Your choice of a Seagate drive was a good one. I have found the Seagate fluid-dynamic drives to be much quieter than Western Digital's ball-bearing drives. I have a WD that sounds like British Airways taking off from Heathrow....
I would go with Fred's suggestion of checking the motherboard settings. There may be something about boot order. There may also be something once you get into the boot record. I do know you are going to have fun.
As for me, I try to keep it simple with only one hard drive connected to the motherboard at the time. I like to keep things as uncomplicated as possible. Good luck!