Uh, actually, no, they do not need to prove you are wrong. If they (the dealership or manufacturer) have reason to believe that the vehicle was not maintained properly, you are the one who must provide the proof that you didn't do something that would void your warranty, such as driving 20,000 miles without a oil change. And, yes, you are correct in saying that anyone can hand write notes where they changed the oil. Those do not stand up in litigation.
We are not in a court of law here. You are not necessarily innocent until proven guilty. You have a fairly strict set of procedures you must adhere to in order to keep your warranty valid. In other words, you must do certain things (like change your oil and filter at specified intervals) for them to do certain things (like honor your warranty). Again, call any manufacturer or dealership, tell them that you have diligently changed your oil filter only, but have run the same oil in your vehicle for 30,000 miles and ask if you still have any warranty on your engine if it breaks. I don't mean to talk down to you on this matter, but let's just say that I have a very inside track on this matter with vehicles.
It has been common subjects for good laughs in break rooms when a service writer tells of how a customer didn't change oil by the manufacturer's guidelines and has their warranty deemed void. They get even a bigger laugh if the customer states that "you'll be hearing from my attorney". Let's just say that in 99 out of 100 of those customers who actually contact an attorney, the attorney will not take the case; unless he is charging by the hour and has no part of his fee based upon a contingency schedule.
I've "been there, done that" for over 20 years, and I've never once seen a customer win when they have neglected to change their oil for 20 or 30 or 40 thousand miles. Regardless of the manufacturer, they all consider that abuse and failure to perform scheduled maintenance. If you have exceeded the time schedule on your Lexus and have an engine failure, I certainly would not volunteer that you have run 20k miles without changing the oil.
I don't doubt that Amsoil, Mobil Delvac, Rotella etc. are great oils, but you will never get a manufacturer pay for a claim if it is known that you have gone well over the prescribed oil change intervals.