Definition of Positive Operator

As stated in Linear Algebra Done Right immediately after the definition of a positive operator, the requirement that $T$ is self-adjoint can be dropped from the definition in the case of a complex inner-product space. However, the self-adjoint condition is needed on real inner-product spaces. Consider, for example, the operator $T$ on $\mathbf{R}^2$ of rotation by $90^\circ$. For this operator $T$ we have $\langle Tv, v \rangle \ge 0$ for all $v \in \mathbf{R}^2$ (because $\langle Tv, v \rangle = 0$ for all $v \in \mathbf{R}^2$), but $T$ is not self-adjoint and $T$ definitely should not be considered to be a positive operator (it has no real eigenvalues).

Your proof is correct in the complex case, which seems to be the case you have. You are correct that you don't need to assume self-adjointness for a complex positive operator (in the real case, knowing $\langle Tx,x \rangle \in \mathbb R$ is not very useful) as it follows from the positiveness

Notice that the conclusion that $\langle Tv,v\rangle= \langle T^*v,v\rangle$ actually implies $T=T^*$ is non-trivial (you can take a look at this question: Proof Complex positive definite => self-adjoint).