As for the question from the title of the article you linked to, the best definition of the term "gerund" that you will find anywhere on the internet is the following: "Gerund is a term that you should forget ever existed in English".
In your sentence the word "singing" functions as the direct object of the verb "make". The word "priority" is a complement of the object "singing".
It leads nowhere to think about verbs as actions, nouns as things etc. The word "singing" does refer to an action, but no less do words like "destruction", "classification" etc. These nouns also derive from verbs, although clearly do not share the suffix with the corresponding verbs destruct, classify etc. There are tons of nouns in English which derive from verbs, and we can make tons more at any given moment.
The key consideration in categorizing words into classes are their distributional properties. So, for example, you won't say "the destruction the city" simply because we distinguish "verbs", as a class of words capable of taking objects, from nouns which cannot do that. It is how we intuitively discriminate between word classes. In the absence of other indicators, we take "singing" to be a noun based on it appearing in a position which is almost reserved for nouns. (not entirely but almost). It is the predicand of the following noun "priority": singing..priority. Or, more generally: She made music her priority.
Depending on how we expand the word "singing" we may think of it as either a noun or a verb (but certainly not as a sort of a mongrel in between these two). Ing nouns and verbs obviously share the suffix, but little more than that. This "little more than that" includes the construction as in your example. Occasionally, a clause headed by an -ing verb can be made an object. So, we won't say that "singing opera" is a noun phrase in:
She made singing opera her priority.
"Singing opera" is an ing clause functioning as the direct object in the structure of this sentence. We don't want to say that "singing" in the original sentence is functionally any different from "singing opera" in the version above. The function is clearly the same, even the form is not. Not only -ing clauses, but also to-infinitival and finite clauses can function as verb object in certain constructions. This is a stance taken by the authors of CGEL and it makes perfect sense.
NOTE: A contributor to the forum says here How is transitivity defined in CGEL? that he "recalls personal discussions with Pullum and Huddleston in which they confirm that only NPs function as objects" . For all I know, this may be true, but in CGEL they took a different position. (which obviously doesn't make a great deal of difference)