Let $(TM,M,\pi,F)$ be the tangent bundle on $M$. I've seen multiple resources state that a vector field on $M$ is a section in $\Gamma_{TM}(M)$ of the tangent bundle. A vector field is a map $$\Phi:M\rightarrow \space ?$$ $$p\mapsto v$$ $$v\in T_pM$$ but a section is a right inverse of $\pi:TM\rightarrow M$, so a section is a map $$\sigma:M\rightarrow TM$$ $$p\mapsto (p,v)$$ $$(p,v)\in TM.$$ $TM=\{(p,v)\vert \space p\in M\space\text {and}\space v\in T_pM\}$, but this can't be the codomain of a vector field because these element's aren't the vectors; they're point-vector ordered pairs.
How do we get around this? And how should vector fields be restated in a similar way?
(Ok, probably I should elaborate a bit more).
You take either one approach:
Either you take the tangent bundle as
$$ TM = \bigcup_{p\in M} T_p M,$$
and then a vector fields will be the mapping given by $p\mapsto v_p$, where $v_p \in T_pM$. Or,
You take the definition of $TM$ as
$$TM = \{ (p,v)\vert \space p\in M\space\text {and}\space v\in T_pM\},$$
for this definition, a vector fields is really a mapping $p\mapsto (p, v_p)$.
Under the mapping $v_p \mapsto (p, v_p)$, as sets these two are the same.
To me the the first definition is much better: the second gives an impression that $M$ is always a subset (the zero section) of $TM$: this is true for tangent bundle, but is false for general fiber bundles.