eligibility criterion for an -ing form to serve as a deverbal adjective

a barking dog / a coughing patient / a sleeping child

I've added the last example as it belongs in the same category as the other two.

I would say that the ing forms here are verb phrases, not adjectives. Even if they were adjectives, they would not be deverbal ones, but adjectives formed by conversion from verbs.

These attributive modifiers are best analysed as verb phrases, not adjectives, for a number of reasons:

They can't be modified by "very".

They can't occur as complement to complex-intransitive verbs like "become" or "seem" (we can't say *"It seemed/became quite barking" or *"He seemed/became quite coughing/ sleeping").

They can't occur as complement to complex-transitive verbs like "find" (we can't say *"I found it quite barking" or *"I found him quite coughing/sleeping").

The range of expressions that can occur as pre-head modifier to a noun is very large and varied: we don't want to call them all adjectives. "Barking", "coughing" and "sleeping" don't have the properties of indisputable adjectives and hence can't belong in that class.