As suggested by the language tag, I think a Veteran Word™ may be
something to do with foreign languages: specifically, a word which is the same in English, French, and German.
All the words in the left-hand column, according to
my knowledge of French, supported by Google Translate for the rarer words and the German,
satisfy this criterion. VETERAN is "only just" a Veteran Word™ because
the French for "veteran" is "vétéran", which is only the same word if we ignore diacritics.
The words in the right-hand column which are "half a Veteran Word™" are so because
the English word is the same in one of French and German but not both: the German for "halt" is "Halt" but the French is "halte"; the French for "assistant" is "assistant" but the German is "Assistent"; the French for "allocation" is "allocation" but the German is "Zuweisung".
They're called Veteran Words™ because
They are 'war veterans' as the 3 countries all fought in WWI and WWII