Learn English – word that describes someone with a “false sense of doing good”

I haven't been able to find a single word that positively describes someone willing to correct others in conversation, but meticulous is a good adjective for someone who cares about details and correctness.

You could combine it with a noun describing what the person is particularly knowledgeable about, e.g. a meticulous grammarian or a meticulous fact-checker.

Moralist reproduces the good denotation of gutmensch with a similar dark connotation:

noun

1.0 A person who teaches or promotes morality.

1.1 A person given to moralizing.
_ODO

Almost everyone considers their own morality to be good. Most consider their moral judgments to be superior, or at least on par with the best, but in the modern mind, a moralist is often portrayed with an irrational moral opinion used unsympathetically to cajole and coerce others into conformity against their will.

John Dewey: An Intellectual Portrait, by Sidney Hook, reveals the positive denotation of one who constructs a superior moral framework:

To those who know him by his less technical writings, John Dewey appears as a great moralist and educator.

In his introduction of The Unity of Plutarch's Work, Anastasios Nikolaidis used moralist with the dark connotations of irrationality and coercion:

These findings, however, do not entail that Plutarch was a crude moralist who stigmatized deeds and conducts, meted out prescriptions for correct ways of living or put forward ideal, and therefore unattainable, patterns of behaviour.

Although the preacher from Nathaniel Hawthorne's The Scarlet Letter was predominantly a hypocrite, he was primarily a moralist, who struggled against his own gutmenschlich qualities at the expense of his secret mistress, Hester Prynne:

The days of the far-off future would toil onward, still with the same burden for her to take up, and bear along with her, but never to fling down; for the accumulating days, and added years, would pile up their misery upon the heap of shame. Throughout them all, giving up her individuality, she would become the general symbol at which the preacher and moralist might point, and in which they might vivify and embody their images of women's frailty and sinful passion. Thus the young and pure would be taught to look at her, with the scarlet letter flaming on her breast—at her, the child of honourable parents—at her, the mother of a babe that would hereafter be a woman—at her, who had once been innocent—as the figure, the body, the reality of sin. And over her grave, the infamy that she must carry thither would be her only monument.
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