Don't pay any attention to US advice for application essays, US university admissions essays are completely different from a UCAS statement (trust me, I've written both successfully).
I wouldn't advise being too casual in your statement, it might work but I've *never* heard of anyone doing that.
You should write about what attracts you to law, definitely. Do write about yourself, show them why you'd be good at law and also what other qualities you have to offer the university. You might want to mention what you're hoping to get out of university, but focus on selling yourself as a hot prospects.
Some of the top universities for Law are UCL, KCL, LSE, Birmingham, Warwick, Nottingham, Edinburgh, Leeds, Glasgow and Bristol - they're fairly different from Oxford and Cambridge and might appeal to you more.
If you check out the Guardian University Guide (see link below), you can also search for rankings according to your predicted UCAS tariff, and check out student satisfaction ratings and university profiles for the places that interest you.