I think part of the problem here is that some Clan Societies confuse the issue here by saying, this is the "COA of Clan X" rather than this is the COA of the "Chief of Clan X".

Terminology is also a big factor in this confusion perpetuated by 'Bucket Shop' Heraldry (no doubt mentioned before) in the use of the word 'Crest' or 'Crest of X' to mean the whole Achievment of arms (Coat of Arms) the Crest, Helm, Mantling and Shield combined.