You did comment that another book struck you as surprisingly advanced, so here's a suggestion that goes in the opposite direction: Elementary linear algebra by Anton and Rorres (and more recent editions seem to have added Kaul as an author). There also seem to be versions with and without applications. I am commenting on the 10th edition "with supplemental applications".
Does have, from your list: vector spaces and transformations, eigenvalues and eigendecompositions, inner product spaces, quadratic forms, as well as the geometric interpretation of these notions. (You might consider the material on inner product spaces a bit thin, perhaps, but it certainly isn't limited to matrices and numerical vectors only.) In the comments you also mention SVD which is covered, but the Courant–Fischer–Weyl min-max principle is only indirectly touched on (when considering the intuition for use of the Rayleigh quotient to find the dominant eigenvalue via the power method). Has lots and lots of questions, not limited to number-crunching — plenty of abstract proof questions too, though they may be a bit short.
Doesn't have: matrix norms, though you'll find them in more specialist texts and I think it would be rare to find a book that covers both introduction to eigenvalues and the norm of a matrix. You may also think that it's short of really challenging questions, but that depends where you're at so is hard for me to comment. (Given that you're preparing for PhD, I suspect you may find the bulk of questions too easy. But my suggestion for the book would only be as a "warm-up" before hitting the harder stuff.)
What you might find it useful for: the topics you're listing sound (to me) comparatively elementary bearing in mind the level of mathematics you intend to study. If it's too big a leap to make in one go, you might use a lower-to-mid undergraduate text to improve your intuition on basic abstract concepts, before transitioning to a heavy-duty text for the more advanced theory.
Why I think you might like it: I know you say you're not so interested in an applications or matrix oriented text, but is it possible that's because "applications" to you has meant "vectors are a tuple of numbers that represent coordinates, matrices represent transformations or changes of coordinate systems" and you want to generalise out from that to something more abstract? This book has loads of applications, many being quite different to that — for example, where the vector space is of polynomials, or continuous functions. If you're from an applied background, I would be very surprised if you didn't find some stuff in there to grab your interest, and I'm pretty sure you won't have seen all the applications before. There's traditional undergraduate matrix number-crunching in there too, but you lose nothing by just skipping over those sections! If you're someone who finds that applications motivate understanding of more abstract concepts, making them more concrete, this might be a really good starter text. Older editions are available cheaply, and despite your protestations to the contrary, I think the "with applications" version may still have some value to you. You probably don't want to go back further than the 10th edition, as this version seems to have beefed up on the theory front.
Why I think you might not like it: if you're at the point in life where you really hate applications, never want to see one again, and just want to be fed raw theory, then this is not your book. Similarly if you want an all-in-one book that gets you straight up to PhD level. You may also decide it's a bit thick for the material you want covered, though I must add that it's a very readable book (there are linear algebra books half the size but written very densely which would probably be a tougher slog to get through than this one).
