It's a good question and I'm not sure there's a clear answer. For example, someone who sees a ghost would probably be said to have had a religious experience, even if they are an atheist who believes ghosts are natural entities.
It is, of course, notoriously difficult to circumscribe a topic like "religious experience". Both "religious" and "experience" are tricky words to pin down. Both Robinson and Beardsworth are aware of this. Beardsworth does not seem particularly bothered by "religious", but he recognizes at least part of the difficulty with "experience": "There is an ambiguity about the phrase 'religious experience', depending on how we take the word 'experience'. If we interpret it on the analogy of phrases like 'interesting experience' or 'harrowing experience', then we shall think in terms of episodes occurring at certain times and in certain places; we shall talk of 'a religious experience' or 'religious experiences'. On the other hand, one can argue, as a contributor did, 'Religious experience is not something to be tied down to definite times and places; it is a way of looking at the world (and oneself) which colours, or should colour, all one's thoughts and actions'" (SP pp. vii-viii).
Robinson refers to the difficulties William James had in finding a definition of "religious experience" even for his own limited purposes, and appears to accept as a positive asset the likelihood that discussion of religion will eventually turn out to be a discussion of everything (TBL p. 5).
For the purposes of the first phase of their research, RERU effectively shelved the difficulty; by appealing to the public at large to submit autobiographical reports, they passed on to their contributors the responsibility for deciding what was to count as "religious experience". But this means that, so far, they cannot really claim to be carrying out very fully the programme outlined by Sir Alister. Their evidence will not, strictly, be evidence of "religious experience", but of what people count as religious experience. An obvious question to investigate now is: why do these people count these experiences as religious? What is the significance of their wanting to use the word "religious"? How are these "religious" experiences related to the rest of their lives, to their social context, to their intellectual beliefs, and so on? Experiences which are phenomenologically similar may serve very different purposes and have very different significance in different situations and for different people. What is "religious" for one person may be non-religious for others. And even if two people have a similar experience and both call it religious, it may still be religious in quite different senses. (Glossolalia, for instance, may be regarded as a way of letting off steam, as a symptom of hysteria, as a religious experience of God, or as a religiously significant experience of demonic possession).
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Faith and Experience V: Religious "Natural History"