How does button[0] know that we are referring to the first button element? Is there some implicit numbering in HTML ?

You're right, a collection of elements (buttons, in this case) will contain them in the order that they occur in the document. So the item with index 0 will be the first element of that type found in the document.

It depends on the context

var mybuttons = document.getElementsByTagName("button") mybuttons[0] // first button

notice the plural "elements", so it returns an array, however something like

var buttonID = document.getElementById("buttonID")

notice the no plural "element" for buttonID,this returns the first match, so the same element would be written as just buttonID ,without referencing an array index

another way of writing it is

document.querySelector("button") //first button

document.querySelectorAll("button") [0] // All buttons so again the first button needs to be referenced as [0]