Whether it's an antitrust suit depends on whether the plaintiff is suing under the antitrust laws, not whether the plaintiff is the government. "Any person who shall be injured in his business or property by reason of anything forbidden in the antitrust laws may sue therefor in any district court of the United States in the district in which the defendant resides": https://www.law.cornell.edu/uscode/text/15/15
Antitrust lawsuits are brought by the government itself, not consumers.
For example, Federal Trade Commission v. Qualcomm Incorporated or United States vs. Microsoft Corporation.