The INSERT operation on a stack is often called PUSH, and the DELETE operation, which does not take an element argument, is often called POP. These names are allusions to physical stacks, such as the spring-loaded stacks of plates used in cafeterias. The order in which plates are popped from the stack is the reverse of the order in which they were pushed onto the stack, since only the top plate is accessible.

You are correct, but as the other reply mentioned, when we're talking about abstract specifications a stack typically doesn't allow reads and writes to arbitrary indices, therefore a stack machine can't perform an in-place swap. The implementation will almost certainly optimize the operation as you're describing, but I'd still expect to see a swap operation defined as "pop pop push push". Likewise I would usually expect to see an add described as popping two arguments and pushing one result, even thought the implementation probably pops one argument, then reads the second and overwrites it with the result.

In short, it's the abstract description of the thing vs the actual implementation.