A growing number of low luminosity and low surface brightness astronomical objects challenge traditional notions of both galaxies and star clusters. To address this, we propose a definition of galaxy independent of the cold dark matter model of the universe: A galaxy is a gravitationally bound set of stars whose properties cannot be explained by a combination of baryons and Newton's laws of gravity. After exploring observational diagnostics of this definition, we examine the classification of ultra-faint dwarfs, globular clusters, ultra-compact dwarfs, and tidal dwarfs. While kinematic studies provide an effective galaxy diagnostic in many regimes, they can be less useful for compact or faint systems. To explore the use of [Fe/H] spread as a complementary diagnostic, we use published spectroscopic [Fe/H] measurements of 16 Milky Way dwarfs and 24 globular clusters to calculate their [Fe/H] spreads and uncertainties. Our principal results are: (i) no old star cluster less luminous then M_V = -10 has a significant (> 0.1 dex) spread in iron; (ii) known ultra-faint dwarfs can be classified with a combination of kinematics and [Fe/H]; (iii) the observed [Fe/H] spreads in massive (> 10^6 M_Sun) globular clusters do not necessarily imply they are the stripped nuclei of dwarfs, nor a need for dark matter; and (iv) if ultra-compact dwarfs reside in dark matter halos akin to those of ultra-faint dwarfs of the same half-light radii, then they will show no clear dynamical signature of dark matter. We suggest measurements that may assist future classification of massive globular clusters, ultra-compact dwarfs, and ultra-faint galaxies. Our galaxy definition is designed to be independent of the details of current observations and models, while our proposed diagnostics can be refined or replaced as our understanding of the universe evolves.

Accepted by AJ; This updated version includes several new references, as well as improvements throughout the text for clarity - in particular clarifying our intended distinction between galaxy "definition" and "diagnostics"