You should check out this thread:
Almost 900 posts, but you’ll find most of the people criticizing Trump and the party that nominated him don’t think much of Hilary Clinton or really all that much of the democrats. Perhaps you are rushing to the conclusion that just because we think Republicans are stupid for selecting Trump means we are Democrats. Quite a few of us aren’t even Americans.
You seem to be very concerned with people painting Trump supporters with a broad brush for someone who is in the #NeverTrump camp. If you feel strongly about Trump being too dangerous/unqualified/whatever to elect, what do you think of people who have elected him?
He isn’t make a historical or logical claim, he is saying that we ought to agree to not recognize the legitimacy of a religion that does call for violence. Stating it as fact is a presumptive close - a sales tactic.
I think Obama’s explanation is good. This is strategic. To whatever extent we associate a conquest-oriented group like ISIS with a religion, we are supporting them by building their pool of recruits and increasing their funding. If one ontological position drives dollars to killers and another avoids doing so, then we aren’t in philosophy class anymore.
Religions are sort of identity driven (and hence we’d be tempted to use self-identification as the test for religious inclusion) but they are also kind of private clubs that have rules of inclusion. That’s obvious when you have a religion with a specific figure head like the pope or the dalai lama who can literally dictate who is in the club and who is not. But there is, to some extent, a consensus model in other religions. If a billion Muslims say that Islam does not condone killing, then a group of people killing in the name of Islam are not Muslims. That’s not a No True Scotsman argument any more than saying Homer isn’t a member of the No Homer’s club is a No True Scotsman. They’ve been excluded for not following the rules.