Go with one of the classics, because they're used for a reason. If you sound too stiff and formal, it comes off like a holiday card. If you reiterate your call to action, it can carry a whiff of desperation. Just get out gracefully and quickly once you've said your piece.
In the bad example, Jean is too vague, and has a few red flags, like suggesting that her last job didn't end well, and making an inappropriate joke about Scorpios to gloss it over. You can tell people you're great, but it's better to prove it by offering specifics, like she does in the better example. In the second example, Jean's letter body is more clearly organized, and makes sure to give the most relevant details about her career so far.