What is the correct behaviour of the "down arrow with horizontal line" key on an iMac keyboard?
It was formerly labeled &quot;end&quot; and should function exactly as you described.
Its function is app-specific. While it's up to the developer to interpret its meaning and implement it in code, exactly what it's supposed to do is not strictly defined anywhere. You can peruse Apple's Human Interface Guidelines but it won't tell you exactly what to do with that particular key, but that document's general principles state user input should always do something the user expects.
Lacking specific guidance it's reasonable to look to the examples Apple provides in their own apps. Using Safari as an example (because I happen to be using it right now) that particular key will scroll to the end of the webpage, no matter how long it might be. In other words, exactly as you expect and not as you experienced in the example you provided. Pressing it again should do nothing assuming the displayed webpage is still at its end. That suggests the app in question is not following Apple's Human Interface Guidelines.
What can you determine from Apple's Human Interface Guidelines? Whereas it does not define that particular key, it provide suggested actions for Down Arrow, Control-Down Arrow, etc. By trial and error I determined that ⌘ Down Arrow performs the same function as the End key—the one you're asking about. I suppose that's relevant if the user doesn't have a full size keyboard as in your picture.
Finally, even Apple occasionally fails to observe their own Human Interface Guidelines. Calling their attention to that fact in a bug report might get them to address such an oversight... eventually.