Note that if the recording devices have microphones, then any method of producing audible sound will also suffice to cause sounds to be recorded. If the sound sounds a lot like a man yelling through a megaphone 500 feet away, does the fact that the megaphoner cannot be seen or identified make it an "unexplainable by science" paranormal event? How about if the sound doesn't sound like anything recognizable, but still comes from a megaphone 500 feet away?
I think there's a good reason why few magic effects are marketed for making electronic devices record sounds. When a tiny chip in a greeting card can record a voice or play a whole song vocals and all, even the most naive audiences today are unlikely to be impressed by an electronic device doing something that electronic devices are known to be able to do. (Indeed, some of us spend a lot of time trying to prevent our computers from doing strange, unexplainable, but decidedly non-paranormal things on a regular basis.)
Thus, using new, unopened, independently purchased devices would be one necessary safeguard. An opaque, sound-resistant, e-m-resistant ("faraday cage") enclosure is also necessary to rule out vibrational and electromagnetic influences. But I think the real missing ingredient here is an objective assessment of whether or not any sounds recorded are causally related to the summoning spell cast by the claimant.
For that I suggest a series of ten trials, each preceded by the either the invocation* or a suitable counter-invocation (asking any paranormal entities to bugger off or at least shut up for a few minutes), selected in random sequence at test time. Each time period to be recorded on a separate tape. Claimant is then given a period of time to listen to the tapes and determine which ones followed invocations and which ones followed counter-invocations. Getting all ten correct constitutes success.
Respectfully,
Myriad
*Note that it takes about 3.5 minutes to recite the invocation in a normal conversational cadence. Reciting it with suitable dramatic effect could easily take twice that. Even The Professor's proposed protocol, that would only leave the paranormal entities 13 minutes to get a word in edgewise; repeating it five or ten times would constitute cruel and unusual treatment for all concerned. Hopefully a more concise invocation could be agreed to. (Doesn't "Here I am; I'm great; Show up!" pretty much cover it? Suggested counter-invocation: "Don't mind me; I'm nothing special; Take a hike.")