The definition of a "miracle" as a violation of the apparent laws of nature, expressed cogently by Hume in his essay on the topic, is common in both philosophy and in the general population. But it's not necessarily the definition used by the majority of religious believers. In C.S. Lewis' Miracles, he explicitly says "Miracles do not, in fact, break the laws of nature." For Lewis, as for many (not all) believers, a "miracle" is an occurrence through which we can perceive the action or the will of God. That could be as dramatic as the transformation of the water to wine, as personal as Augustine hearing "take and read" in the garden, or as a much a shared everyday experience as a beautiful sunset.
As you have noted, this means the miracle inheres in the perception of the believer, and not in the event itself. This is "a feature, not a bug." For the Christian believer, at least, the personal nature of the relationship with God is a foundational belief. From that point of view, the idea of an objective, depersonalized, scientific taxonomy of miracles is arguably point-missing. The miracle is an expression of God's personal love--a form of communication. It has a sender and a receiver. It can not be adequately understood in isolation from either.
With all that said, there are also many believers--and religious institutions--who do have a conception of miracles closer to that outlined in the original post. I find that less personally defensible, and will pass over it in silence.