If the target creature is no longer on the battlefield when Reflector Mage resolves, the target creature's owner is still prevented from casting spells with the same name, correct?
No, they are not. The only thing preventing them from casting a spell with that name is the ability when it resolves. And, if the target is illegal when it tries to resolve then it doesn't. None of its effects happen.
Note, that Reflector Mage itself doesn't target anything, it is its triggered ability that targets. And that ability only goes on the stack after the Mage has already resolved and entered the battlefield. The target for the ability is chosen at that time. The ability then still has to resolve. So players can respond while the Mage is already on the field, but the ability has not yet done anything.
The scenario as described is a shortcut proposal, since the target for the ability is prematurely announced. Removing the intended target before the Mage resolves allows its controller to choose a different target when it finally does resolve and enter the battlefield.
The scenario as described is a shortcut proposal, since the target for the ability is prematurely announced.
Ah, right... I remember reading that in some cases the target is not (need not be) announced immediately. So in this case, because it's a triggered ability, the target need only be announced after the trigger takes place, is that it?
Ah, right... I remember reading that in some cases the target is not (need not be) announced immediately. So in this case, because it's a triggered ability, the target need only be announced after the trigger takes place, is that it?
Reflector Mage itself doesn't target anything; with limited exceptions, no permanent spell can have targets (C.R. 115.1b, 115.1e). Rather, Reflector Mage's enters-the-battlefield ability has a target, namely a "creature an opponent controls" (C.R. 115.1d). In general, if you cast Reflector Mage and choose such a creature in advance, you will be bound to that choice unless another player intervenes (C.R. 722.2a-c; for sanctioned tournaments, see M.T.R. 4.2). See also this thread, which discusses a similar situation involving an ability of the form "As [this creature] enters the battlefield...".
EDIT (Apr. 17, 2020): Edited to conform to rule update with Ikoria: Lair of Behemoths.
I remember reading that in some cases the target is not (need not be) announced immediately. So in this case, because it's a triggered ability, the target need only be announced after the trigger takes place, is that it?
Yes. Since you are the one playing Reflector Mage, it's important that you actually wait for your opponents to respond (or to indicate that they choose not to), before announcing your target. The usual way of doing so is to cast Reflector Mage and then ask "does it resolve?".
Someone may hold an Essence Scatter in their hand but may be hesitant to spend it on your spell, not knowing what you'll actually target.
The scenario as described is a shortcut proposal, since the target for the ability is prematurely announced. Removing the intended target before the Mage resolves allows its controller to choose a different target when it finally does resolve and enter the battlefield.
Former Rules Advisor
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EDIT (Apr. 17, 2020): Edited to conform to rule update with Ikoria: Lair of Behemoths.
Someone may hold an Essence Scatter in their hand but may be hesitant to spend it on your spell, not knowing what you'll actually target.
RULES OF MAGIC :
http://magic.wizards.com/en/game-info/gameplay/rules-and-formats/rules