Or would this one work?
Nope, that’s a regular HDD, not an SSD. 🙁 The specs say: “Storage Drive Type: HDD”
The title should clearly say “SSD” if that’s what it is. Another clue is when you look in the specs and it says “5400 rpm”: that’s a sign of a (spinning) platter disk, not a solid state disk.
The first one you linked to is more like what you want. I’ve never used that brand before, (I’ve always used Samsung or Crucial SSDs) but it looks like reviews aren’t too bad. Samsung has the best reviews, but they’re $20-30 more. Looks like Crucial is only $9 more.
I just realized you could go with something like this and keep your 1 TB hard drive in for (slower) storage as well! (This would replace the 128 GB SSD drive that comes with it, and leave the 1 TB SATA drive in place.) They’re more expensive per GB than the SATA SSD drives, but they can theoretically give you a lot faster performance. It’s up to you. 🙂