Is the covariant derivative of a vector field (not the components of a vector) same as the partial derivative? I am adding a screenshot from page 69 from General Relativity: An introduction for physicists by Hobson where they used the partial derivative operator $\partial_bv$ for vector field derivatives, where in other books the covariant derivative operator $\nabla_bv$ is used.
Relevant: is a vector operating on another like $u(v)$ (as in Lie Bracket) also mean covariant derivative of v in the direction of u?
Your book uses an old point view in differential geometry, where "the covariant derivative of a vector field" $\nabla_i v^j$ is just a coordinate expression which equals the components of "the partial derivative of a vector field", so that $$\partial_iv = \sum_j(\nabla_iv^j)e_j.$$ This is also the point of view taken in Sokolnikoff's tensor calculus and in Grifeld Tensor calculus and the calculus of moving surfaces, but it is kind of outdated.
In the modern point of view, the "partial derivative of a vector field" (i.e. $\partial_i v$) is not defined, but the same object is called the covariant derivative of $v$ and is denoted $\nabla_i v$, so that its component decomposition is $$\nabla_iv = \sum_j(\nabla_iv^j)e_j.$$