No, not really. What's the universal generalization or group here, of which Damore is a counter-example? "Googlers who have written memos like Damore's and were then fired"? Damore's situation is a specific one, and he made specific claims about the importance of freedom of expression at Google:

A. Damore spoke out about how Googlers shouldn't feel unsafe to express themselves.

B. People at Google are reportedly intimidated about expressing themselves.

C. We would expect Damore to be unhappy that Googlers are feeling unsafe to express themselves.

That's not a logical fallacy. Doesn't mean that idleword's rhetorical argument can't be criticized for being overbearing. Or for being wrong in its premises, e.g. Damore never explicitly said freedom of expression at Google was important, he alleged that Google management's policies stifle ideological viewpoints. That doesn't mean Damore has an opinion about Google employees being intimidated by the public.

I see what you're saying, but it's an edge case for the No True Scotsman, with a mix of "moving the goal posts," i.e what real sincerity is like.

>No, not really. What's the universal generalization or group here, of which Damore is a counter-example?

"Sincere people."

There's also tinges of "wishful thinking."

Don't get me wrong, the rhetorical trick that idlewords is implied to use is just as annoying. Like, "If pro-life people truly cared about human and babies lives, why don't they speak up for single-payer healthcare?" It involves irrational assumptions and/or over-simplification of issues. And its implications are itself a logical fallacy: If people who support A also support B, and people who support C support B, then people who believe in A but not C are hypocrites

But No True Scotsman fallacies can occur without those pre-requisites and it's a common enough fallacy that it deserves its own term, so that it's easier to identify and denounce its use.

Also, I really want to know if there is a specific phrase/category that describes what idlewords is accused of, for the same reasons of easier identification and denunciation :)

You're right, and I was wrong before - the example of "If X was a true Y, they would do what I say all Ys should do" seems like it might be related to No True Scotsman but it's definitely different. It seems like the fallacy would be in the speaker's definition of "a true Y", so maybe it's using a "Persuasive definition"[0]? I'm not certain, but I'm also curious if there's a better term!

[0] https://en.wikipedia.org/wiki/Persuasive_definition