When Twitter Inc. earlier this month announced a new fact-checking tool, it was billed largely as a measure to combat false information about coronavirus. Two weeks later, the company deployed that tool in one of the biggest actions in its history: squaring off with President Trump.

Since Tuesday, the platform has taken several actions on messages from Mr. Trump as well as a post from the official White House account, marking some as breaking the company’s rules and adding a fact-check label to two about mail-in ballots. In…

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