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Smattering said:But still, in this particular case, the statements "at ##r=0## there is a singularity" and "the domain of the metric does not include ##r=0##" and "the metric is not well-defined at ##r=0##" are equivalent?
No. The most fundamental reason that such a definition doesn't work is that the singularity is not a point or point-set in the spacetime manifold. Therefore it doesn't make sense to ask whether the metric is defined "there." There's "no there there" at which we could even ask whether the metric is defined. Specifying ##r=0## does not specify a set of points in the manifold; those values of the ##r## coordinate are not part of the coordinate chart.
Note that the metric could also be undefined at a certain coordinate value without there being any singularity. For example, when we express the metric of the Schwarzschild spacetime in the Schwarzschild coordinates, it's undefined at the event horizon, but this is not considered a singularity. (People describe it as a coordinate singularity, but that just means not a real singularity.)