Everyone seems to use the slope definition of stability for vector bundles without making any mention to the fact that this should be the correct definition describing that a stable equivalence class of vector bundles is then equivalent to a stable orbit under the action of the gauge group on say, the space of all connections.

Edit:

To be more precise, The notion of a stable orbit of a group action requires that, say, the stabilizer of an element be discrete. Now, viewing vector bundles as isomorphism classes under the action of the gauge group, one says that a vector bundle is stable if it satisfies the slope condition (all sub-bundles have strictly less slope). The question is, how is this slope inequality equivalent to the definition of stable orbit for group actions.