> You get too close to a star or planet, you will also fall in
An object with a relatively much smaller mass can take a hyperbolic orbit arbitrarily close to a star or planet without "falling in". Practical examples include <https://en.wikipedia.org/wiki/List_of_hyperbolic_comets> and many small near-earth objects. Theoretical details in Newtonian gravity at <https://en.wikipedia.org/wiki/Hyperbolic_trajectory> and there is a literature exploring post-Newtonian corrections for such "orbits" (in e.g. General (or Numerical) Relativity, or formalisms like gravitoelectromagnetism or effective one body, for cases where one or both bodies are "compact" (like white dwarfs or neutron stars), the mass-ratio of the bodies is close to 1, or one or both bodies are moving at large fractions of c, or there is some combination of these features).
An object can take a hyperbolic orbit arbitrarily close to a black hole without falling in.
Black holes were first formalized in the context of General Relativity; commonly one would use a different term to describe something phenomenologically similar but set in a different theory ("fuzzballs", gravastars, and so on). Fevers and spots appear in diseases with very different causal agents (any number of quite different bacteria, viruses, and other things may cause grossly similar symptoms in the victim). Likewise, apparent trapping surfaces can appear in many ways in different theories of gravitation or in different configurations of variables within a single theory of gravitation.
The important word in the previous sentence is "trapping": anything crossing that surface from the outside to the inside cannot cross back to the outside for arbitrarily long times. Differences in configurations on the inside of a trapping surface do not materially affect the outside at all. The "apparent" qualifier captures the possibility that the trapping surface does not go to the eternal future because of (for example) instabilities from quantum effects (Hawking-style), so we can take at all as meaning "for a very very very very verrrrrrrrry long time".
The observables of a black hole in General Relativity (in its form as a physical theory that adequately represents many physical features of our universe) are all outside the event horizon. Anything inside the horizon stays permanently inside. From the outside one cannot test the internal configuration. While the internal configuration is described in several black hole solutions to the Einstein Field Equations, nobody expects that just because the external configuration (outside the black hole) is a good physical model, that therefore the internal configuration must be a good physical model too. Roy Kerr makes this point almost every time he lectures about his solution for black holes with nonzero angular momentum (example: 48m04s mark <https://youtu.be/nypav68tq8Q?t=2884>, where he points out that the Kerr solution is a vacuum solution, and that adding matter inside the horizons is likely to dramatically change the black hole internal configuration. Note however that adding matter to the outside part of the Kerr solution is highly likely to be undramatic, and that is one reason why the Kerr solution is astrophysically useful).
Stars and planets differ from black holes in that there is no apparent trapping surface. You can shoot an electron neutrino right through the Earth or the Sun. You can't shoot an electron neutrino through a black hole: if it goes in, it stays trapped inside.
Your term "singularity shadows" presupposes that as we develop better solutions (numerical or analytical) of the Einstein Field Equations for (apparent) astrophysical black holes, the singularities that appear in e.g. the Schwarzschild or Kerr solutions will remain. That may not be true. I don't think the "shadow" part adds any accuracy.
It is not true that "gravity always implies matter". In General Relativity there exist several exact solutions to the Einstein Field Equations where there is significant spacetime curvature but no mass. Some of these usefully approximate features of the universe we observe, even though as far as we can tell there is no part of our universe that is completely free of matter (in the most general sense, including electromagnetic radiation), although large and growing regions are so sparse that the matter in them does not collapse gravitationally into clumps. This trend is just as much an effect of spacetime curvature as is the gravitational collapse of dust clouds into stars.
Finally, you can use whatever nomenclature makes you happy. It's just a fanciful term that covers a wide range of theoretical descriptions and astrophysical phenomena. Astrophysicists and theorists use "black hole" knowing that they may be talking about quantitatively and qualitatively different objects with fairly similar symptoms being presented. But they also know how to find, read, and understand a precise mathematical description that removes the ambiguities of English (and other languages) and any inaccuracies (in some settings a black hole may be a very weak greybody radiator; and in some settings "hole" may be less poetic and more descriptive, e.g. in Wheeler's bag-of-gold solution). Substituting some other pithy name for "black hole" doesn't help these physicists, and is unlikely to help anyone who doesn't know how to deal with the formal, unambiguous descriptions of them.