Is there a word that means to smoke but not to inhale?
I often hear this referred to as "puffing," as in "Puffing on a cigar," or "Puffing a cigar." However, "puffing" is still vague enough that it doesn't always convey that the smoke isn't being inhaled to the lungs. More often than not, it's just slang for smoking.
I have heard "bum puffing" for this. Although that has a negative connotation, as in it is smoking 'for show' or not really smoking.