You're thinking about this wrong. Tunnel says "When you discard this, you may reveal it." By the time you reveal it, you've already answered the question of where it got discarded from. You just have to ask, "Did I discard this Tunnel?" Yes? Then you can reveal it.
Beggar actually allows you to discard it. Since it doesn't specify a source, it has to be from your hand. Otherwise, Hamlet would allow you to discard cards in play to gain an action or a buy.
Hamlet is not relevant, since it's not talking about a specific card. The only way Beggar could allow you to discard it from anywhere, would be because it refers to a specific card (meaning a specific copy of a card).
Anyway, I'm convinced that Tunnel is not relevant to the question either. (That's what I was getting at in the last sentence in my previous post.) So I agree with you there. We can just compare with "gain". Watchtower ("when you gain a card") triggers when you gain from anywhere, even though it doesn't refer to a specific card. So the reason Tunnel triggers when you discard from anywhere, is not that it refers to a specific card.
So then Beggar and Market Square are entirely unique. Discarding and gaining are the only two card movements that have a default from-location. And there are no other cards that tell you to discard or gain a specific card without telling you where to discard it or gain it from.
I guess the way I was thinking, was with regards to lose-track. Although in this case it's not about losing track. Lose-track tells you that a movement effect tries to move a card from where it's expected to be. Now, when you discard Beggar from your hand, that's where it expected itself to be of course. But doesn't that apply when it's on your Island mat too? Of course, the argument that you're making is that plain "discard" actually means "discard from your hand", so the other stuff doesn't matter. Look at Loan though: "Reveal cards from your deck until you reveal a Treasure. Discard it or trash it. Discard the other cards." Discard doesn't mean from your hand there; it's referring to cards with a known location, so that overrides the default. My way of thinking was that a specific copy of a card always knows its own location.
I'm not at all convinces that that's correct anymore, but I don't see that it's so clear-cut either.