That is an interesting take. Perhaps there are those who a reserve particular subject matter for the name science. Is it possible to give a date when this shift in the meaning occurred?
The view that only science is science is redundant. The view that areas of human experience cannot come under the scrutiny of human knowledge is the very monopoly of a dogmatism that you warn against.
Theology has never been a science.
I can understand why you mention my statement is dogmatic in nature. Actually, I'm basing it on a strict definition of science which is: a branch of study in which facts are observed and classified, and, usually, quantitative laws are formulated and verified; involves the application of mathematical reasoning and data analysis to natural phenomena (from the McGraw-Hill Dictionary of Scientific and Technical Terms). In other words, scientific understanding is a subset of understanding.
Your statement that "areas of human experience cannot come under the scrutiny of human knowledge" does not fit into the classical definition of science. Theology is one such example. Can the mind God (let alone the mind of a human) be reduced to a series of mathematical equations? Could the thoughts of a poet be expressed as a formula? Can the Bible be deconstructed in a lab, a theory developed and new Bible verses predicted?
My point is that thee are many areas of human understanding that fall outside the realm of science. Science is only one way of building understanding. It is systematic and cold in its ways. I do not demean theological study when I say it is not science, because the nature of theology is not scientific, nor is science theological. But that does not mean that either are diminished in the realm of human understanding. They are different tools that let us understand different things.