I define it by one of two ways:
1. The duck rule: if it looks like a duck, quacks like a duck, and walks like a duck, it is a duck.
2. The sovereign can do whatever he wants rule: if the legislature says it is a coat of arms, then it is.
The latter need not be expressed in the statute adopting the arms. That statute may well say, "The seal of the State of Oblivia shall consist of a shield, Argent charged with a porcupine between three kumquats proper; crest a sun rising Azure; supporters two escaped convicts in chains; all surrounded by a ring bearing the words....", etc. What we have here meets test 1; if other statutes and official acts then refer to this design as "the arms of the state," then it also meets test 2.
The act adopting the Colorado state seal, for example, does not say, "these are the arms of Colorado" in so many words. But to me, at least, the intent is clear: "An heraldic shield, bearing in chief..." and so on. And even clearer in light of other early statutes referring to it as "the arms of the state," including the laws authorizing seals for the state supreme court, adjutant general, and bureau of mines, as well as an act of April 9, 1907, providing "that a state banner be and the same is hereby adopted to be used on all occasions when the state is officially and publicly represented. Said banner shall consist of the state coat of arms upon a dark blue ground."
Unfortunately, rule 2 requires me to accept that when Texas says that its coat of arms is the design on its seal, minus the encircling rings, lettering, and any background, it means this actually
is Texas's coat of arms: