Why not deism?

[Inqwizitor]

My level of trust would be based on the claim. 

If the claim is mundane: "I went to the grocery store today" I may take them at their word. There may be a receipt or video evidence but why would I give a shit? 

I expect reality to be consistent. The dead do not return to life. Humans do not regenerate lost limbs. Horses do not fly. If someone were to make such a claim I would expect extraordinary evidence. I would also be unable to verify that evidence so would insist the claimant take their claim to experts and I would await their report. As far as subjective religious experiences, cornball anecdotes hold little weight with me. Their god can come impress me personally if it chooses. It would know what would convince me.

So if a bunch of your friends said they saw a horse flying around, you would say, no way, not unless I personally saddle up that flying horse, will I believe.

It's certainly not an unreasonable position. It would only get to be adamant and obstinate if they then take you to the flying horse, which you see circling the treetops, and then you assume it must be some sort of hallucination, because that sort of thing just doesn't happen.

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No different. I would hope that the expert witnesses would support their argument with scientific data. I would hope the expert witness was not taken at their word just because they are considered an authority.

If you did not personally observe and verify that scientific data, then you are trusting them at their word. Judges are forced to do this sort of thing often, and yes they are sometimes led astray, and sometimes with tragic consequences.