Several months later and the machine is finally running from an SSD.
The machine also got slower and slower with the HDD that was already installed:
I opened the lid and it took a very long time before I could finally type in my password. Sometimes it took so incredibly long the screen went to black again before I was even able to type a single character in the password field. Wake it up again and wait a bit longer until I could finally type in my password. Press return after typing the password and wait again until it finally starts loading the desktop.
Open Safari and when it's finally loaded open a webpage. Do the two finger gesture to scroll and it would show the beachball for about 30 seconds before it would finally start scrolling. Just typing an email address took time, because every time I press the two keys to type the at-sign the machine had to think for about 30 seconds before it'd finally found the character corresponding to the two-key combination.
A FaceTime call came in and it took about two minutes to start the FaceTime app. The icon in the Dock was just jumping up and down while the system was busy trying to start FaceTime.
These are just some examples. It was just slow as **** and as discussed earlier in this thread I and some other posters were suspecting the SATA cable. I ordered a new one and today I finally replaced it. Then I put the HDD back and everything seemed fine. I made a Time Machine back-up and put the SSD in the machine. It took less than 45 minutes to restore all data on the SSD, it restarted and everything is running fine now. Everything is faster and smoother than it has ever been and System Report now also shows the machine is finally running on SATA III speeds (6 Gbps).
So, if everyone experiences these kind of problems, try replacing the SATA cable. From what I've seen on the internet this really is a problem with these models.