I purchased an adapter which connects my MacBook Air (2013) SSD to PCI-E of motherboard.
However, I was unable to find a proper way to read it, even with MacDrive.
What I want to do is back up the files off of this SSD as my MacBook Air has been damaged and unable to boot.
Is there any recommended solution?
Mac drives are formatted as HFS filesystem. You need software like Disk Internals Linux Reader to read the partitions. However you will be unable to write on the disk in its format.
To make the disk work again in Mac you'll need to make a bootable USB drive with Mac OS system (maybe BootCamp) to format the drive and install system.
You might be running into an issue with Apple's CoreStorage volumes. CoreStorage became the default format when reinstalling OS X as of Mavericks, and Yosemite outright converted your system to it during upgrades.
CoreStorage volumes are to Apple what Dynamic Disks are to Windows -- a proprietary container format for your partitions. The underlying file system is still HFS+, but the HFS+ volume is wrapped in a CoreStorage partition that few Windows-based HFS+ drivers understand. MacDisk does support them, but only if you are running version 9.3.2.6 or later.
Check your version of MacDisk. It looks from the screenshot above that it doesn't understand what (partition2) is. It is very likely a CoreStorage volume.
