Your not alone here in being confused with SATA
While the dialog of SATA is the same across all of the current versions, the I/O speed is not. You have the first generation of SATA (SATA I) which ran at 1.5Gb/s then soon afterwards SATA II 3.0Gb/s and now SATA III which runs at 6.0Gb/s.
Over the course of time the HD vendors have pushed forward with faster I/O drives but the system makers have lagged a bit. In addition the older systems as you have guessed here may not be able to handle these newer HD's. Some HD/SSD vendors have offered drives which auto-sense (mostly SATA II/III). Most offered a jumper to select an older compatibility mode (1.5Gb/s). To add more confusion some drives don't offer any choice other than SATA III.
What to do?
First you need to find out what your system is capable in supporting. In your case you might have a SATA I or an undocumented SATA II HD interface. Make sure you have updated the EFI firmware (to enable SATA II if your systems chipset offers it). Here's the Apple TN on how to check it and update it EFI & SMC firmware updates for Intel-based Macs. Depending on your version of Mac OS-X you can use Profiler or the About Mac to review your systems hardware to see what SATA speed your system has.
You will need to review the HD specs to make sure it offers SATA I or SATA II I/O speed your system is limited to. Many so called auto sensing drives have had problems. So I would speak with someone from the HD vendor to see if they have tested the given drive out on your model. And download the document from there web site on how to set any needed jumper.
Make sure to ask for the required jumper for your HD (there are a few sizes) and get a few as they are small and very easy to loose!
As to height you must not exceed 9.5mm. The current 2TB drives will not fit and some of the older 1TB drives where also too high.
Hopefully that answers your questions - Good Luck!