For Elimination, that card is unlocked by playing the stand-alone Elimination playlist (not Trials of Osiris). It has been available once so far, so just keep an eye out and hopefully it will rotate back in soon. Your Trials of Osiris round wins are counting towards the tiers on this card. The Double's card does not have game win tiers (Double's wins that are not an objective game type are counted by the Skirmish card). I don't know if the Salvage card will have your already earned wins recorded on it when it unlocks.