Spoiler alert: a letter of intent is not the same as a cover letter. They're similar (being letters and all, and focused on yourself), but are actually used in different situations. Your cover letter is what you write when you're applying to a specific job you found through traditional channels (online job search, referral, recruiter). It details why you're a great fit for this particular job. A letter of intent is what you write when you're cold-calling (leaving a resume without being solicited for one), or applying for a job in a more general situation, like a job fair or submitting your resume to a general pool. The letter of intent is similar in that you're selling yourself, but tends to be less granular about a particular position. Letters of intent are often more networking-related, or aspirational, than position-oriented. Because you may have fewer specifics in hand about what you're applying for and who will be reading your application package, it's likely you won't have the most personalized opener. That's okay! Be general, but professional, formal, and polite.
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.