The definition of $\oplus$ I would like to understand why the books give two different concepts to $\oplus$ between vector spaces:

See:

Concept 1: $W=V_1\oplus V_2=\{(v_1,v_2) \mid v_1\in V_1, v_2\in V_2\}$.

Concept 2: $W=V_1\oplus V_2=\{v_1+v_2 \mid v_1\in V_1, v_2\in V_2\}$, where $V_1\cap V_2=\{0\}$.

Are they equivalent?

I'm thinking to prove this is an isomorphism $(v_1,v_2)\mapsto v_1+v_2$.

Thanks
The two concepts are indistinguishable from the point of view of algebra.

The mathematical formulation of this is:
$$
V_1\cap V_2 = \{0\}\implies
\{(v_1 +v_2 |v_1 ∈V_1 ,v_2 ∈V_2 \} \cong\{
(v_1,v_2) |v_1 ∈V_1 ,v_2 ∈V_2 
\}
$$

I let you find the isomorphism.
In addition to MattS's excellent answer, we can add that the two notions, Cartesian products and external direct sum, differ when considering infinitely many vectors spaces.

We consider a family $(V_j)_{j\in J}$ of vector spaces, indexed by a set $J$. Then: 


The product $\displaystyle{\prod_{j\in J}V_j}$ is, as a set, the Cartesian product of the $V_j$ with the following operations: if $v=(v_j)_{j\in J}$ and $w=(w_j)_{j\in J}$, $\lambda v=(\lambda v_j)_{j\in J}$ and $v+w=(v_j+w_j)_{j\in J}$. It is easily checked that these two operations give to $\displaystyle{\prod_{j\in J}V_j}$ a structure of vector spaces.
The external direct sum $\displaystyle{\bigoplus_{j\in J}V_j}$ is the subspace of $\displaystyle{\prod_{j\in J}V_j}$ of all vectors $v=(v_j)_{j\in J}$ such that all component $v_j$, except a finite number of them, are zero.


For a finite number of spaces $V_j$, these two spaces are the same. 

These definition are not actually restricted to vector spaces but can be applied to many algebraic structures (groups, modules, rings,...), with the suitable operations.

Edit: after re-reading the OP and the comments, I feel this answer is more a complement to Pedro Arbizu's comment and Mookid's answer.