By the definition, if $W$ is a vector space and $U_1,U_2$ are the subspaces of $W$ then the direct sum $W=U_1 \oplus U_2$ holds iff the following is true: $W=U_1+U_2$ and every vector in $W$ has a unique representation as a sum of a vector from $U_1$ and a vector from $U_2$.
However, I read here that "by the definition, the condition for a sum to be direct is $\mathbf{u_1}+\mathbf{u_2}=\mathbf{0},~~ \mathbf{u_1} \in U_1, ~\mathbf{u_2} \in U_2$"
and I fail to understand why is that true. The definition says that $\mathbf{u_1}+\mathbf{u_2} \in W$, but why $\mathbf{u_1}+\mathbf{u_2}$ is necessarily equal to the zero element? What is the point then of the direct sum if every sum of two vectors from two subspaces is zero?
Let's suppose $W=U_1\oplus U_2$ in the sense you have defined in your first paragraph.
Note that $$ \underbrace{0}_{\in W}=\underbrace{0}_{\in U_1}+\underbrace{0}_{\in U_2} $$ so by the uniqueness of the representation on the RHS above, if you have $$ \underbrace{0}_{\in W}=\underbrace{u_1}_{\in U_1}+\underbrace{u_2}_{\in U_2} $$ then $u_1=0$ and $u_2=0$.