Is Hearthstone a game-within-game?

Hearthstone is set within the Warcraft universe (so no Sarah Kerrigan card will be incoming), but that is about the extent of the known "lore" about the game.

While there are Hearthstone items in World of Warcraft itself, there's no way to tell if the intent was for it to be an actual, playable game, or merely Blizzard self-referential humor.

If anything, Hearthstone's tutorial implies that the heroes are actually playing against each other (as in, you're not playing Jaina, you are Jaina), which makes no sense within the warcraft universe, simply because half of the class heroes are canonically dead.

TV Tropes would call Hearthstone a Gaiden Game to World of Warcraft:

A game which is related to another game by taking place in the same universe, but has completely unrelated plot and game mechanics.

Can you play hearthstone within WoW or WoW within hearthstone? There is your answer. Pretty much each game has a story but that does not make it a game within a game. Take day of the tentacle where you could play inside a computer within that game another game called `maniac mansion', now that is a game within a game. Or sid meiers pirates that holds many mini games would count as games within a game.

Simply putting it like you do would be wrong. Like already being mentioned, lore is being used to give the game a theme. Playing a random Starwars game would not mean you are actually being an actor, knowing mr Lucas and playing a game together with Jabba the Hutt. You are just playing a game with the Starwars lore.