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In case of Kherson and Kharkiv they used the word counter-offensive because it has a use in propaganda, it further emphases in the reader (Western audience) that they justly defending themselves and that any measure they take it isn't aggressive, attacking, but it's only self-defense. The Russians attacked Ukraine, so they can't tell Westerners that they are attacking too. It's manipulation, that's how they paint their narrative. They don't care if it's wrong in military sense, the average reader can't tell anyway.
Western media is very proficient in telling facts in a way that it is completely twisted in order to establish narrative and influence the way people think about the conflict.
Colonel Reisner used the word Operational right however.