This is going to seem like a really stupid question as a basic understanding of the English language will answer it, but to clarify:

If a card, such as True Mage, says 'deal X indirect damage to each player' then I assume I am correct in that both the controlling player of the card and his opponent both take the damage when the action is triggered?

Exactly. If a card doesn't limit the players or units affected, it really does mean all of them. So True Mage and Loremaster of Hoeth deal indirect damage to each player, not just the opponent, and Troll Vomit destroys all units, including the ones of the player playing it.