The definition of direct sums and subspaces?
The sum of two subspaces of a vector space $V_1,V_2\subseteq V$ (denoted as $V_1+V_2$ is defined as
$$V_1 + V_2 = \{v_1 + v_2| v_1\in V_1\land v_2\in V_2\}$$
and it is a vector subspace of $V$.
The direct sum of two vector spaces $V_1, V_2$, (denoted as $V_1\oplus V_2$) is a vector space $V_1\times V_2$ with operations defined as you wrote.
The connection between a direct and ordinary sum is that if $V_1, V_2\subset V$ and $V_1$ and $V_2$ are linearly independent, then $V_1+V_2$ is isomorphic to $V_1\oplus V_2$.
The $+$ is defined on pairs of subspaces of a given vector space and $\times$ on pairs of vector spaces (not considered as subspaces of any sorts) and we write $V_1 \oplus V_2$ in place of $V_1 + V_2$, if $V_1 \cap V_2 = \{0\}$.
These two things are "the same" in the sense, that if $V_1,V_2$ are subvectorspaces of $V$ with $V_1 \cap V_2 = \{0\}$, then $V_1 \oplus V_2$ is isomorphic to $V_1 \times V_2$ which is not hard to prove.
I have just found this pdf: http://www.math.ncku.edu.tw/~fjmliou/advcal/sumvspace.pdf which summarizes direct sums, and explains what is going on here. I will here outline what it says. Thanks also to the other answers which helped clarify things.
This is to do with the difference between an internal and external direct product:
The External Direct Sum
Let $V$ and $W$ be two vector spaces over $F$ then we the external direct product $V\oplus_e W$ is defined as I have given it in the question. i.e. the vector space with underlying set $V\times W$ and operators defined by: \[(u_1,u_2)+(v_1,v_2)=(u_1+v_1,u_2+v_2)\] \[c(v_1,v_2)=(cv_1,cv_2)\]
The Internal Direct Sum
For $V_1$ and $V_2$ subspaces of $V$ which satisfy:
- for each $v\in V$ there exists $v_1 \in V_1$ and $v_2 \in V_2$ such that $v=v_1+v_2$.
- $V_1 \cap V_2\{0\}$
in this case we write $V=V_1 \oplus_i V_2$ and $V$ is the inner direct product of $V_1$ and $V_2$.
Link between them
There is a linear isomorphism such that $V_1\oplus_e V_2$ is isomorphic to $V_1 \oplus_i V_2$