Until end of turn, target player can't
castTo cast a spell is to take it from the zone it’s in (usually the hand), put it on the stack, and pay its costs, so that it will eventually resolve and have its effect. A player may cast a spell if they have priority. See rule 601, “Casting Spells.”
To cast a card is to cast it as a spell. (CR 701.5) instant or sorcery spells, and that player can't
activateTo activate an activated ability is to put it onto the stack and pay its costs, so that it will eventually resolve and have its effect. Only an object’s controller (or its owner, if it doesn’t have a controller) can activate its activated ability unless the object specifically says otherwise. A player may activate an ability if they have priority. See rule 602, “Activating Activated Abilities.” (CR 701.2) abilities that aren't mana abilities.
Draw a card.