You have the correct parts listed here the carrier and a HD that will fit the carrier.
From the writeup this appears to be the Seagate 2.5" SSHD (hybrid) that works with SATA I (1.5 Gb/s), II (3.0 Gb/s) & III (6.0 Gb/s) so it will work just fine in your system. Here's the spec sheet from Seagate: Laptop SSHD
Frankly, I would recommend upgrading the main HD to a Seagate 3.5" SSHD. Review this: Seagate Desktop SSHD & here is the Spec sheet for the drive series Desktop SSHD This drive will work with your system without any problems (don't forget to take your thermal sensor off your old drive). Doing this will give you better performance and will be less work.
If you don't want to upgrade your current drive then I would go with a straight SSD Vs a SSHD for the second drive.
If you can try to get an external FireWire HD case to hold your old drive and make upgrading a breeze.
Apple uses a few different sources for their HD's (Seagate, Toshiba & Western Digital).
The issue is more of the SATA I/O speed (SATA I, II or III) the system has and what the given HD you want to use is able to work in your system. The next issue is the given size of the drive (2.5" laptop class vs 3.5" desktop class) as well as the drives height if the space is limited and in laptops. You might encounter one issue which is the power connector/thermal sensor cable may need to be swapped if it has an integrated sensor Vs an external sensor or you are trying a different vendors drive from what the cable was designed to work with. As it turns out its not a big deal to swap out the cable to the other type.
If your system is still able to boot up. You can find out what HD you have via the About This Mac under the Apple menu, then clicking on More Info button then click the System Report button a new window will pop up with the report expand the hardware twisty and click on Storage. Under Physical Drive: copy the Media Name and do a google search for it it should tell you what the make and model of the drive you currently have in your system.
You can use EveryMac using your serial number you should be able to get more details of your system. Sadly, it won't tell you who's HD you have. You will need to open the system to find out if its dead.
The debate of Seagate Vs WD is pointless both make good drives and both have been stung with a bad run of drives. I would focus on what features and specs a given drive has that you are looking for. We use both as replacements depending on what we are looking for.
Given your primary drive is dead, I would take the quicker and easier path just replace it with the SSHD.