Re: Another possible explanation for the differences
"All the truthful tweets are written by someone other than Trump. So not detecting true/false so much as Trump/NotTrump"
Surprisingly, this doesn't seem to be something the boffins have considered, but is (IMO) quite likely.
It's certainly safe to say that a lot of his tweets are written by others - usually the more official/diplomatic ones, such as the recent tweet regarding the discovery of that lost Argentine submarine.
Even with just a brief skim through Trump's twitter feed (and I seriously wouldn't recommend spending more than 5-10 minutes there), you will find his followers attempting to identify tweets written by others. This is genrally based on a combination of spelling, puctuation (e.g. excessive use of "!"s), word use and tone. When they were tweeted can also be a factor, e.g. his early-morning burst of tweets while watching "FOX and Friends" is a regular source of hysteria and misinformation...