A participial adjective (or adjective participle) is, according to OED, "an adjective that is a participle in origin and form". In other words, think of a participle as being a word with two possible functions. Whether you call it a 'participle' or a participial adjective is largely a matter of the angle from which you look at it - the emphasis you want to lay on its function in your analysis.
- Where it is found with an auxiliary verb, it makes a verb phrase. Its function is to carry the meaning of the phrase, where the auxiliary communicates such attributes as the voice, aspect and tense.
- When it is found with no auxiliary, it is sometimes best thought of as an adjective, largely without a verbal function. In such cases, and to make clear that it is not being used as a verb, it is convenient to call it 'a [participial] adjective'.
Consider the phrase 'a thinking person'. Here the word 'thinking' is clearly formed from the verb 'to think' with the '-ing' inflection which AWE calls 'the -ing participle. But in this case, the word is not being used as a verb: it is merely describing what sort of person we are talking about. A 'considered opinion' is a particular sort of opinion - one that has been thought about. One guide to whether a given word is better thought of as a participle or an adjective is to consider whether any modifier or intensifier such as 'well-', 'very' or 'not quite' can be applied to it. If it can, then the word may best be thought of as an adjective.