I've never really questioned this before. I've always accepted push ups as the term one uses when pushing his body up and down, Lol. I think both press up and push up is self explanatory to be honest. If you use any one of those terms in a conversation, the person will instinctively know what you're talking about. 

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Haha I'm not surprised the only time we ever had a debate on exercises was when people were kipping and claiming it counted as a pull-up... It's called a pull-up not a flap-around-like-a-fish-outta-water lol

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carlosfandango

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My time in the military they called them press ups, because it is a pressing motion. But push ups works in civvie street because well... You are pushing up...  But I guess the press part of it comes from the motion similar to a chest press/bench press. Like you mentioned they're pretty interchangeable.