It is common in physics literature to identify invariant geometric objects with their components in some reference frame.
For example, if $T$ is a type (2,0) tensor field, then it is common to refer to its components $T^{\mu\nu}$ as a tensor field.
In essence, the connection coefficients are the components of a connection. So it is also customary to refer to $\Gamma^\rho_{\mu\nu}$ or $\omega^{\ a}_{\mu\ b}$ as a connection.
Do note that the exact definition of the object themselves are subject to convention. There are authors who refer to different things as a connection, than the $\nabla$ covariant derivative. In fact, there are general connections that do not admit a representation as a covariant derivative. In general, a connection is always a rule to define parallel transport of locally defined geometric objects along curves. But the exact way of defining them differ from author to author.
Also, the Levi-Civita connection and the "spin connection" are essentially the same. The Christoffel symbols $\Gamma^\rho_{\mu\nu}$ are the components of $\nabla$ when taken with respect to a holonomic (coordinate-) frame, while the "spin connection" coefficients $\omega^{\ a}_{\mu\ b}$ are the components of $\nabla$ when taken with respect to an orthonormal frame.
The only reason why there is any significant difference between the two is that spinor fields can only be represented in orthonormal frames, so to define covariant derivatives of spinors, you need the "spin connection". But this is all just juggling with local component representations, the underlying invariant object is the same.
