I think you are quite right, and this adapter would do the job you want. That is, allow you to use a cable with a Slim SATA connector on one end to connect to a normal SATA connector set on your new SSD unit. What threw me was that the DESCRIPTION of the adapter on that eBay site says it is used for the opposite application. It says it would allow you to connect from a mobo/PSU set of standard SATA connectors to a device with a Slim SATA input, like a laptop optical drive. But the PHOTO posted on that site shows that the Slim SATA end of it would ACCEPT a Slim SATA input cable connecor such as your cable has. In fact, according to the photo, that adapter would NOT be able to plug into a Slim SATA socket on a laptop CD unit, and could not accept standard SATA cable ends.
Just a note of caution here. The eBay web page thus contradicts itself, because what it shows in the photo does NOT match what it describes. That means it is in error. So, which error? Is the photo correct and the description wrong, so that the actual unit WILL do your task? Or, is the photo posted the wrong one, and the actual adapter being sold is according to the descriptive words? I encourage you to check that detail before buying.