"Files can be attached to your email at the push of <a> <the> button." — What is the rule for choosing an article here?

Question

ldoceonline.com:

at/with the push of a button:
(1) Files can be attached to your email at the push of a button.

As far as I understand, "a button" means "one of the buttons", i.e. some button non-specific yet. Therefore, "a button" isn't appropriate for (1) because the button to attach files which is talked about is the only and specific one. That is, there must be "the button" in (1) rather than "a button".

Given this, why is (1) correct?

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my variant:
(2) Files can be attached to your email at the push of the button.

Is (2) correct?
If it is, then what is the difference between (1) and (2)?
If not, then why not?

Answer 1

The collocation template “P the N₁ of a N₂”—where P is a preposition and N₁ is a motion, gesture, or other action—is simply idiomatic. Thus, besides at the push of a button, we also have for example,

• at the flick of a switch,
• at the drop of a hat,
• in the blink of an eye, and
• with the twist of a knob.

In all of these, the emphasis is on the ease with which something can be made to happen or the simplicity of its cause. And for that purpose it’s not as important to specify exactly which button, switch, hat, etc. is involved.

At the push of the button isn’t syntactically forbidden. It simply sounds wrong to native speakers. If one’s context does require specificity about which button, then one could rework the sentence to something like All it takes to attach files to your email is pushing the button.

Answer 2

If you use "the" in that sentence, it is not understood by the audience which button you're talking about. In many cases, the person saying this doesn't know which button it is, as in:

You could approve my application with the push of a button.

I'm thinking of a standard keyboard here, where hitting the "Enter" button might be the key that attaches files to your emails, but that button has several other functions, and there are probably other buttons that do the same thing, like clicking a checkbox or "OK" button with the mouse.

It's possible to use "the" to specify a certain button, but it has to be explicit:

You could approve my application with the push of the button in the lower-right corner of your screen.

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Note: I also agree with Paul's answer (+1 from me) that the structure used may be "simply idiomatic". My answer doesn't explain why we can't use "the" and have the listener simply understand that we obviously mean the button that does the thing. I'm writing this answer to show what "the" would mean to a native speaker, which is a different way of explaining why not to use it.