1. Current definition

The ad consequentiam argument scheme is currently defined as follows:

(1) Appeal to consequences, also known as argumentum ad consequentiam (Latin for « argument to the consequence »), is an argument that concludes a hypothesis (typically a belief) to be either true or false based on whether the premise leads to desirable or undesirable consequences. (Wikipedia, Appeal to consequences).
God must exist: if he does not exist, then very many people pray for nothing!
(Wikipedia, Argumentum ad consequentiam, in French)

Walton (1999) traces the “Historical Origins of Argument Ad Consequentiam”.

2. Different meanings of ad consequentiam

The Latin word consequentia means:

1. 1. “What comes after” in space or in time.
   2. The logical consequence: per consequentias, “it follows that” (Gaffiot, Consequentia)
      Lat. ex quo natura consequi ut… “from which it is a natural consequence that” [1]

In the first meaning, an argument ad consequentiam refers to something that happened after a central event. For example, a large sum of money was stolen from Paul. The investigator notes that after the date of the robbery, James, an acquaintance of Paul’s, spent large sums of money, while nothing changed in his income. The investigator uses  an argument based upon « what happened after (the theft) » to charge James with the theft, S. Circumstances.

In the second meaning, an argument ad consequentiam is an argument based on causal or logical consequences. With this meaning, the label ad consequentiam covers all the range of appeals to consequences (effect-to-cause arguments), be they positive or negative:
— The pragmatic argument appealing to positive consequences is an ad consequentiam argument.
— In the same way, appeals to absurdity are a form of refutative appeal to consequences deemed absurd a) from a logical point of view (they lead to a contradiction), or b) undesirable from a psychological or moral point of view, or c) contrary to the pragmatic interests and values of the speaker.

So, one may fear that definition (1) misleadingly reduces the various effect/consequence to cause arguments to what can more conveniently be called pathetic argumentation, if we are to judge by the example.

[1] « Again, [the Stoics] hold that the universe is governed by divine will; it is a city or state of which both men and gods are members, and each one of us is a part of this universe; from which it is a natural consequence that = ex quo natura consequi ut we should prefer the common advantage to our own. 
(Cicero, De Finibus, Bk 3) https://penelope.uchicago.edu/Thayer/E/Roman/Texts/Cicero/de_Finibus/home.html]