Interior Sphere Condition

Your reproduction of the example in Gilbarg-Trudinger is not quite correct: $\log x+iy$ should be $\log(x+iy)$. This is how I pictured the region $\{x:\operatorname{Re}z>0, \operatorname{Re}(x/\log z)<0\}$ in Maple:

It's quite close to losing smoothness at the origin. This is how it works: since $\log z=\log |z|+i\arg z$, the argument of $\log z$ is $O(1/\log |z|)$ near the origin. Therefore, the effect of the denominator is to rotate the line $\operatorname{Re}z=0$ by the angle about $1/\log |z|$. This means that the curve looks like $x=|y/\log x|$ near $x=0$. The derivative is continuous, but with merely logarithmic modulus of continuity.

The reason G&T use this particular example is to demonstrate how the Hopf lemma fails in this domain; the function $u=\operatorname{Re}(x/\log z)$ is harmonic and has maximum at the origin, yet the normal derivative vanishes there.

If one is interested just in showing that not every $C^1$ domain satisfies the interior sphere condition, it's easier to consider $\{(x,y):y>\psi(x)\}$ where $\psi\in C^1$, $\psi'(0)=0$, and $$\lim_{x\to 0} \frac{\psi(x)}{x^2}=\infty \tag{1}$$ For example, $\psi(x)=|x|^{3/2}$ or $\psi(x)=|x^2\log x|$; the latter function belongs to $C^{1,\alpha}$ for all $\alpha<1$. Condition (1) makes it impossible to touch $(0,0)$ by a disk from inside the domain, since the equation of the boundary of any such disk would have $y=O(x^2)$.