Mine Workers v Bagwell (1994):

Although the procedural contours of the two forms of contempt are well established, the distinguishing characteristics of civil versus criminal contempts are somewhat less clear. In the leading early case addressing this issue in the context of imprisonment, Gompers v. Bucks Stove & Range Co., the Court emphasized that whether a contempt is civil or criminal turns on the “character and purpose” of the sanction involved. Thus, a contempt sanction is considered civil if it “is remedial, and for the benefit of the complainant. But if it is for criminal contempt the sentence is punitive, to vindicate the authority of the court.” Ibid.

As Gompers recognized, however, the stated purposes of a contempt sanction alone cannot be determinative. “[W]hen a court imposes fines and punishments on a contemnor, it is not only vindicating its legal authority to enter the initial court order, but it also is seeking to give effect to the law’s purpose of modifying the contemnor’s behavior to conform to the terms required in the order.” Hicks. Most contempt sanctions, like most criminal punishments, to some extent punish a prior offense as well as coerce an offender’s future obedience. The Hicks Court accordingly held that conclusions about the civil or criminal nature of a contempt sanction are properly drawn, not from “the subjective intent of a State’s laws and its courts,” but “from an examination of the character of the relief itself.”

(Footnotes and pincites omitted.)

That case goes on to cite United States v. United Mine Workers of America (1947) for the proposition:

This dichotomy between coercive and punitive imprisonment has been extended to the fine context. A contempt fine accordingly is considered civil and remedial if it either “coerce[s] the defendant into compliance with the court’s order, [or] . . . compensate[s] the complainant for losses sustained.”

Certainly the judge may impose a fine to coerce Twitter’s obedience. The judge may also impose a fine to compensate injured parties for Twitter’s disobedience?as long as that compensatory fine is calibrated to the harm suffered by those injured parties and payable to them.

But when a fine is calibrated instead to Twitter’s ability to pay, and levied for the purpose of upholding the court’s dignity, then the character of the contempt proceeding changes.