Actually, Market Square is the only one that says "from your hand."
"Discard", yes. But all Reaction cards (except Tunnel) tell you to do something "from your hand". Lots and lots of other cards also says "from your hand". Explorer is one example.
It seems like for Market Square there could be some confusion IF that clause weren't there - it means that if you play a bunch of market squares and then a chapel, you can't discard the market square from play to gain gold.
But is there that confusion with beggar? I mean, where else would you discard it from? It can't be in play because it's not your turn, you can't really discard things from the discard pile?
Why is the clause there on Moat, Horse Traders, Fool's Gold, etc? Because if Moat had said just "reveal this", you could reveal it from anywhere - your discard pile, Supply, Trash, a mat... There's no rule that says you can't reveal or move cards that are anywhere, as long as they're not lost track of at least. There are cards that let you do things when it's not your turn, there's a card that lets you reveal cards in your discard pile (Tunnel), there are cards that let you move things out of Trash. Etc.
When it comes to discarding, it's true it often doesn't say "from your hand", because that's the default. But those instructions are telling you to discard other cards. Without specifying where those cards are, the default is that they're in your hand. Beggar on the other hand is telling you to discard itself, and we know where it is. If it happens to not be in your hand at the point when it triggers, it's still saying you may discard it (because it's missing the clause).
Treasury
_ +1 Card, +1 Action, +1 Coin, When you discard this from play, if you didn't buy a Victory card this turn, you may put this on top of your deck.
Herbalist
_ +1 Buy, +1 Coin, When you discard this from play, you may put one of your Treasures from play on top of your deck.
Alchemist
_ +2 Cards, +1 Action, When you discard this from play, you may put this on top of your deck if you have a Potion in play.
Tunnel
_ 2 VP // When you discard this other than during a Clean-up phase, you may reveal it. If you do, gain a Gold.
Beggar
_ Gain 3 Coppers, putting them into your hand. // When another player plays an Attack card, you may discard this. If you do, gain two Silvers, putting one on top of your deck.
Hermit
_ Look through your discard pile. You may trash a card from your discard pile or hand that is not a Treasure. Gain a card costing up to 3 Coins. // When you discard this from play, if you did not buy any cards this turn, trash this and gain a Madman from the Madman pile.
Market Square
_ +1 Card, +1 Action, +1 Buy // When one of your cards is trashed, you may discard this from your hand. If you do, gain a Gold.
Treasury: Discard triggers from play. Nothing happens on opponents' turns.
Herbalist: Discard triggers from play. Nothing happens on opponents' turns.
Alchemist: Discard triggers from play. Nothing happens on opponents' turns.
Tunnel: Discard (by default) from hand. This card cannot be in play. Discard can happen on your turn or opponents' turns.
Beggar: Discard (by default) from hand. This card only triggers the discard during opponents' turns. Nothing happens on your turn.
Hermit: Discard triggers from play. Nothing happens on opponents' turns.
Market Square: Discard (by default) from hand. This card can be in play. Discard can happen on your turn or opponents' turns.
Market Square is the weird one because it can be in play when one of your cards is trashed. It has "from your hand" because it
could be interpreted wrongly as part of the "while in play" text.
Moat
_ +2 Cards // When another player plays an Attack card, you may reveal this from your hand. If you do, you are unaffected by that Attack.
Secret Chamber
_ Discard any number of cards. +1 Coin per card discarded. // When another player plays an Attack card, you may reveal this from your hand. If you do, +2 cards, then put 2 cards from your hand on top of your deck.
Watchtower
_ Draw until you have 6 cards in hand. // When you gain a card, you may reveal this from your hand. If you do, either trash that card, or put it on top of your deck.
Trader
_ Trash a card from your hand. Gain a number of Silvers equal to its cost in coins. // When you would gain a card, you may reveal this from your hand. If you do, instead, gain a silver.
Moat: Reveal only happens on an opponents' turn.
Secret Chamber: Reveal only happens on an opponents' turn.
Watchtower: Reveal can happen on your turn or an opponents' turn. "From your hand" means it cannot be in play.
Trader: Reveal can happen on your turn or an opponents' turn. "From your hand" means it cannot be in play.
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I don't think there's much confusion here about these rules. I mean, I can't just point at Chapel on turn 1 and say "That says I can trash up to 4 cards from my hand" and proceed to trash 3 estates and a copper.