I have a buddy who's pet peeve is when people sound out the last word in an abbreviation. NIC Card instead or SAM missile. I also find this an acceptable mistake.
The simple definition for arcade is " a place with many games that can be played by putting coins in them"
Someone saying "Oh you have an arcade game" is technically perfect, but I don't mind "Oh, you have an arcade". I have 5 games. Not sure if that would qualify as "many games", but since they are mame, they play 100's of games, which definitely qualifies as "many games" but I don't have to put coins in them, so if we are going to be technical I guess it would never be an arcade? This is confusing. I'll just stick with being ok with either Arcade or Arcade Game.