That is simply not how the language is used, in practice. If the DM tells you that an enemy can be detected on the far end of the room, then you should take that to mean your character detects an enemy on the far end of the room. It's like asking if you can borrow a pencil; it's not technically the correct question to be asking, but it's how the language works in practice.
As far as the rules are concerned, the Invisible condition tells you the degree to which it makes you hidden: "for the purpose of hiding, the creature is heavily obscured." Is a heavily obscured character automatically hidden - "both unseen and unheard"? No, of course not. The rules tell us that you can still be heard as normal, and also tell us that it requires a Stealth check in order try and hide. The Hiding rules tell us that an invisible character can always try and hide, not that they are often or generally considered to be hidden by default unless environmental circumstances suggest otherwise.
Put simply, your interpretation is not what is intended by the rules. There are too many places where the rules would have been written differently if your interpretation was intended, and while you're welcome to change any rules at your own table, you should be aware of those changes so that you can convey them to your players and prevent misunderstandings.