You said that providential interventions should not be understood “solely as intervention on the purely horizontal plane”. How can anything be understood as an intervention ‘from above’ on a ‘purely horizontal plane’ in the first place? Providential intervention denotes not merely horizontal, materialistic causality, hence the contradiction, unless you mean something different by “horizontal plane” than I am inferring you to mean.
A man in a boat said something during a storm. The storm stopped. There is a materialistic causality between words spoken and the weather? What natural laws were broken? None, but it was providential timing and placing to the extreme, the same M.O. as other cases noted in ‘objective evidence’ linked above, the same sovereign God not “divorced from His eternal being” and while fulfilling his eternal decree.