Cathair and *cair are in fact two distinct Scottish Gaelic words which have been confused over a long period in ways not yet fully understand.
Cathair ultimately derives from Greek kathedra via Latin cathedra. This fell together with a separate Irish / Old Gaelic word cathair, meaning ‘chair’. This also exists within a British (i.e. Pictish/Cumbric) context and occurs in place-names to mean a variously a fort, a hill, or in some cases a chair-shaped prominence. These elements and their relationship with each other are not yet fully understood.
*Cair is in origin either a native Celtic term meaning ‘enclosure’ or is ultimately a loan word form Latin quadra, ‘a square’. This term appears throughout Britain, and originally in Scotland within a British context. It often applies to the site of Roman fortification such as Cramond, Kirkintilloch and Kirkcaldy. As Gaelic replaced Cumbric and Pictish in these areas, the term would have been adopted as Gaelic *cair. It was perhaps adopted into Scots as keir, meaning a fort or mound.
Whilst there certainly some names in Scotland which derive from cathair, many of the Car- names, especially in the South and East of Scotland surely relate to the British term *cair. Despite the similarity of the words *cair and cathair in both pronunciation and usage, they are in fact distinct. The confusion between the two forms is a recent understanding introduced by popular place-name publications.