Tywin is just the game. I hate to harp on this too, but the dude playing Joff is amazingly good in this role - there's a lot going on there. There seems to have been a series of great Tywin to family interaction in recent episodes. Called Cersi stupid, called Tyrion a disgrace to his house (though still of his house), and told Joff he could have him carried to meetings.
The only early introduction I saw was that of Tywin, when he was cutting up game, which I watched because it was so amazingly effective and interesting (and set a precedence for the characters always doing something while talking - which is why I thought he was going to whip Joff in the throne room a couple of weeks ago) because was also a scene that was originally so similar to the book debut of Samwell Tarly's father (who we will meet later, if he hasn't appeared yet on the TV show, I don't know).
The whole straight up MAN/father cutting up meat intro was just class though, and so appropriate for the head of the most powerful family.
and it wasn't just any kind of game, it was a stag! haha!
and yeah, that's tywin's first appearance. charles dance could be the nicest man in real life, but he always scared me a little bit. ever since 'last action hero' when he had the interchangeable fake eye...
and with regards to jaime, you're right that the show SHOULD have cemented jaime being a bad ass. other than the fight with ned in season 1, i don't think i've seen him cut loose. well, he does kill his cousin and karstark's son but that was while he was chained.
basically, his rep in the show is mostly inferred from the plot and the dialogue. (people saying "jaime was blah-blah-blah" etc.) and it might not be obvious to the casual viewer other than being "important person from an important family". which, for the purposes of the TV show, is good enough IMHO.
and that's probably a thing with bringing ASoIaF to TV: we don't get all the nuances of POV characters in the novels. in the book, a character's adventure is riveting stuff. in TV form, non-readers complain its boring. a lot of inner monologues and detail is lost.