As your source indicates, "absolute" has different meanings. "Ablative absolutes" had a very well defined meaning in Classical Latin, and traditional grammarians of English borrowed the term for somewhat similar constructions in English though there was nothing "ablative" about them. Your source also indicates that many modern linguists do not use the term "absolute."
I do not find your source's usage compelling. As far as I am concerned, the "absolute" phrase in your source's main example technically modifies "storks" rather than every aspect of the independent clause. To be sure, the phrase supplements the information in the independent clause, but "their bodies" directly relates to storks rather a singular orange sky.
Having rejected your source's usage as not helpful, I must agree with your analysis. "A crime punishable ..." is an appositive supplementing the noun phrase headed by the gerund "plotting."