When does $\|x+y\|=\|x\|+\|y\|?$

Let $(V_\mathbb R,\langle,\rangle)$ be an inner product space. I’m trying to see for $x,y\in V$ when does $\|x+y\|=\|x\|+\|y\|?$

Where did I go wrong?

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The third line in your formulas can be written (if you use that $\langle x,y\rangle=\langle y,x\rangle$) as $\langle x+y,x+y\rangle$. So you have equality in the inequality in your second line. This implies
\langle x,y\rangle=\|x\|\,\|y\|,
i.e. equality in Cauchy-Schwarz. So $x$ and $y$ are colinear.

Regarding your mistake, the two sides of the inequality in your first “implies” are equal. So your second “implies” is wrong.

Suppose $||x+y|| = ||x|| + ||y||$. Then we have
\langle x + y, x + y \rangle &= \langle x, x + y \rangle + \langle y, x + y \rangle\\
&= \langle x, x \rangle + 2\langle x, y \rangle + \langle y, y \rangle\\
||x+y||^2&= ||x||^2 + 2\langle x, y \rangle + ||y||^2\\

But from the assumption we have $||x+y||^2 = (||x|| + ||y||)^2 = ||x||^2 + 2||x||\,||y|| + ||y||^2\\$. By the Cauchy–Schwarz inequality, this means x and y are linearly dependent, and specifically $x=cy$ for $c\ge 0$ (notice the absolute value around the inner product in the Cauchy-Schwarz inequality).

I just realized I never addressed your question. The first implication is incorrect, since $\langle x+y,x+y\rangle-\langle x,x+y\rangle=\langle y,x+y\rangle$. In a real inner product space, your “inequality” is an equality, and in a complex vector space it is not safe to assume inequality.