I'm Lucy, and I'd be happy to answer your questions today.
The relevant statute here is Title 27, Code of Federal Regulations, Section 478.11. Title 27 governs ATF affairs. There is nothing in this statute that suggests a person who has been ordered to take anger management classes has been involuntarily committed to a mental institution. The word "anger" does not appear anywhere in that statute. Here are the two definitions that could be relevant.
Adjudicated as a mental defective. (a) A determination by a court, board, commission, or other lawful authority that a person, as a result of marked subnormal intelligence, or mental illness, incompetency, condition, or disease:
(1) Is a danger to himself or to others; or
(2) Lacks the mental capacity to contract or manage his own affairs.
(b) The term shall include—
(1) A finding of insanity by a court in a criminal case; and
(2) Those persons found incompetent to stand trial or found not guilty by reason of lack of mental responsibility pursuant to articles 50a and 72b of the Uniform Code of Military Justice, 10 U.S.C. 850a, 876b.
Committed to a mental institution. A formal commitment of a person to a mental institution by a court, board, commission, or other lawful authority. The term includes a commitment to a mental institution involuntarily. The term includes commitment for mental defectiveness or mental illness. It also includes commitments for other reasons, such as for drug use. The term does not include a person in a mental institution for observation or a voluntary admission to a mental institution.
There was some discussion back in 2014 that the ATF might want to expand this definition in the way you're saying, but the current definitions do not say anything about a person who has been ordered to take anger management being prohibited from owning a firearm.