Both Uses

acquit
in
The Mill on the Floss
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But a man who had had a university education could teach anything he liked; especially a man like Stelling, who had made a speech at a Mudport dinner on a political occasion, and had acquitted himself so well that it was generally remarked, this son-in-law of Timpson's was a sharp fellow.
Chpt 1.3 *

acquitted = handled (conducted or behaved)
It seemed a wrong toward him that his uncle Deane had no confidence in him,—did not see at once that he should acquit himself well, which Tom himself was as certain of as of the daylight.†
Chpt 3.5 *

Definitions:

(1) (acquit as in: she was acquitted)
to officially find "not guilty" of criminal charges

Being acquitted does not mean the court declared the person innocent. It simply means there was not enough evidence to prove guilt according to the law.

The amount of evidence required depends on the type of case. In a criminal case, where someone could go to prison, the prosecution must prove guilt beyond a reasonable doubt—a very high standard. In a civil case, a lawsuit usually about money, the plaintiff only needs to show that the defendant is responsible more likely than not—a much lower standard.

This is why the same person can be found "not guilty" in a criminal trial (their guilt was not proved beyond a reasonable doubt) but still be found guilty in a civil trial for the same actions (because their guilt was more likely than not).
(2) (acquit as in: she acquitted herself well)
to handle oneself in a specified way -- which is typically in a positive way
(3) (meaning too rare to warrant focus)
Much less commonly and archaically, acquit can have other meanings. It can mean to release from a duty, as when Jane Austen wrote "I cannot acquit him of that duty" in her novel, Pride and Prejudice.

It can also mean to perform or complete an obligation, as when Charles Dickens wrote "I have a business charge to acquit myself of," in his novel, A Tale of Two Cities.