I'm a bit confused about the definition of a subbundle when it comes to its status as a submanifold.
Take $\pi: E \to M$ a smooth fiber bundle.
When $E$ is a vector bundle, Lee's Intro. to Smooth Manifolds defines a vector subbundle to be, among other things, an embedded submanifold. But I'm currently studying more general and principal bundles and this point is generating some confusion, since different authors mean different things by submanifolds. My two main sources are Kobayashi-Nomizu (Foundations of Differential Geometry vol I) and Poor (Differential Geometric Structures).
Now Kobayashi calls an imbedded submanifold what for Lee is an immersed submanifold. Poor never defines what a submanifold is on his text, but it cites Warner's Foundations of Differentiable manifolds as its main source on general manifold theory, which in turn defines a submanifold to be an immersed one by Lee's definition.
This confused me when was working with the problems of showing a given subset is a subbundle of a more general bundle. Usually, showing immersion is a straightforward process, but then I'm not sure I if also have to work on embedding. For example, I was able to show the pullback bundle is immersed by straightforward computations, but the steps given here show that the pullback bundle is actually embedded, and this last step is fairly elaborated.
So, what is the standard definition (if there is one) I should take for subbundles on a more general context?