Regarding this question,
I have figured out what is meant by spectral projection, and have thus found the answer as well. A spectral projection is the image of $x$ under a step/indicator function defined on its spectrum, which is hence an orthogonal projection on some closed subspace. (Here I am using the Bounded Borel functional calculus.)
Okay, it's a projection. And I said the word "spectrum" in the above paragraph. But surely it's more "spectral" than that. Does the image of the so-obtained projection have anything to do with $x$ of a spectral nature? (This is an open-ended question, so please qualify what sort of spectral significance there is. But some specific interpretations are below):
Suppose $\lambda$ is an eigenvalue of $x$ which hence means that its in the spectrum of $x$. Does the indicator function of the one point $\lambda$ applied to x via the Borel Functional Calculus project onto the eigenspace associated to it? By considering $z*\chi_\lambda$ one can see that the image of this projection is a subset of the eigenspace of $\lambda$. Does the other inclusion hold?
More generally, if $\lambda$ is only a general element of the spectrum, is the indicator function of it applied to $x$ equal to a projection projecting onto some sort of "spectrally significant space for $\lambda$?"