Besides the interpretations you mention, you can think of the normalizing constant as the value of the prior predictive distribution at the observed x. If the prior predictive is discrete then this is a probability mass, and if the prior predictive is continuous it is a probability density.
The prior predictive is in the continuous case is $$ p(x) = \int_\Theta p(\theta)p(x|\theta) $$
Which is a distribution that assigns probability mass/density to the outcomes in the sample space. Then when x is observed it is fixed at the observed x and fits in the denominator of Bayes' theorem.
However, note that with continuous distributions there is no mathematical constraint on the density value assigned to a set with measure zero (i.e., zero probability), and since any specific point on a continuous distribution indeed has measure zero then technically the value of the density on the prior predictive at exactly x can be set arbitrarily. But that aside, I think this way of visualizing the normalizing constant is fairly intuitive.
You can read more here. (Let me know if you don't have access) This too, which is a bit more modern.