I'm sorry to say Seagate did us in! This version of drive won't work in your system.
The previous generation of the Seagate SSHD had auto sense technology to match up the SATA speed with the given systems SATA port. You see, your system is only a SATA II (3.0 Gb/s) but the current version of Seagate's drive is a fixed SATA III (6.0 Gb/s) drive as they needed to cut the cost of the drive to be competitive.
Note the differences in SATA Transfer Rates Supported (Gb/s) line. You'll need to either find one of the older models or jump to a SSD that has auto sense support like the Samsung 860 EVO 2.5" SSD
The HD SATA cable often needs replacing in these older systems. As a point of process I always replace it when I do a drive replacement as I've seen to many come back. Here's the IFIXIT guide you'll need to follow: MacBook Pro 13" Unibody Mid 2010 Hard Drive Cable Replacement and I use the newer version of the cable which is this one: MacBook Pro 13" Unibody (Mid 2012) Hard Drive Cable You'll need to modify the anchor point a bit to get it to fit. I also place electricians tape on the uppercase where the cable crosses over as the rough surface of the aluminum case abrades the cable causing issues as well and don't over bend the bends as the wires inside are fragile.
As for the OS install its self, you really need to setup an external OS installer drive. Follow this guide to setup a USB thumb drive for the task: How to create a bootable macOS High Sierra installer drive. Let the installer configure the partitions as it wants to create the hidden boot partition. If you want to make a second partition adjust the partition after you've installed the OS.