The whole arcade/game/cabinet differential could also be a regional thing, not just a generational issue. I'm from England and back in the day, all kids I knew including myself would generalise the term 'arcade' to refer to both the venue and the machines themselves. It usually depended on the context in which it was used, but it was mainly because it was simply a familiar word to abbreviate to in general conversation or when pestering your parents in anyway on the subject!
Again there are other Brits who will not have done this, but I'm just talking from my experiences. These days I do tend to be more specific from being on boards like this, but then in real life I've had to adopt the very American term 'video arcade machine' or simply 'pac-man machines' when talking about what I do for a living as all definition of the correct term has also been lost over time, as if I say 'arcade cabinet' they think I build grabber machines and if I say 'arcade games' they always assume fruit machines (another debatable term as we grew up referring to those as 'bandits').
To be honest, I just don't care how like-minded people care to refer to arcade machines, so long as they're comfortable with it. I'm not the kind of guy who walks round telling people "it's nuc-lee-er not nuc-ular!" or "for god's sake, it's Porsch-er!"