>>OTOH, Trump's doctor's letter is clearly a fake.

Based on an opinion piece by a journalist?!

My observations would be
a) The grammatical error in the salutation is surprising. I'd expect a letter for this purpose to be checked and rechecked by the author then given an editorial once-over by colleagues before being vetted by political advisors.
b) The trumpeting about "positive" findings equating to disease, isn't that interesting. The physician refers to a complete medical examination, not just "tests" with positive results as per this reporter's paraphrase. It's unusual, but the context I read is that his health is good.
c) A medical letter is an opinion, so if the physician is prepared to use extravagant language, that's their opinion. It's not addressed to a colleague and I'd expect his language was tailored for the intended audience- but I'm not competent to second guess his reasoning. Nor is a journalist. Clearly the purpose of the letter is to say that Trump has no health issues.
d) People criticize a general medical opinion from a gastroenterologist. However, outside the US a gastroenterologist also is a general medical specialist which also happens in the US, so I'd expect to see more about his career before finding fault.
e) The website error is embarrassing: it appears that the domain name was grabbed via a Chinese domain business in April and now redirects inappropriately. I'd expect the domain now is held to ransom unless a big fee is paid to acquire it- unless it was secured by a HRC supporter in which case it'll redirect to malware sites or porn until the election. Usually you can claw back a neglected domain name for up to a year, so it seems this one had to be left hanging for a while.
f) The letter header still has his dead father. This suggests the letter header is old and needs to be updated which suggests administrative carelessness- .but why was there no editorial input here?!
g) I don't attach meaning to the inclusion of an email address. He knew what the letter was for and that reporters and others would invade his practice by other means unless he provided a contact. A smart person would provide a gmail address that presumably can be abandoned in due course.

Unusual and partly embarrassing, but fake? Not convinced- if anything I'd say that it's more authentic, lacking even rudimentary political shaping to head off obvious gaffes.