Equivalent notion of approximate differentiability Is it true that the definition of approximate differentiability presented here of a function $f: \mathbb{R}^N \to \mathbb{R}$ is equivalent to the following one?

$$\lim_{r \to 0} \rlap{-}\!\!\int_{B_r(x)} \min \left\{\frac{f(y)-f(x) - L(y-x)}{|y-x|},1 \right\} dy = 0$$
  for some linear $L:\mathbb{R}^N \to \mathbb{R}$.

 A: I think the answer is no.
First of all consider a related question: is the standard definition of approximate continuity of $g\colon \mathbb R \to \mathbb R$ equivalent to the following one
$$
\lim_{r \to 0} \rlap{-}\!\!\int_{B_r(x)} \min \left\{|g(y)-g(x)|,1 \right\} dy = 0.
$$
This definition is in fact strictly weaker. Consider for instance the function $g(x) = \sum_{n=1}^\infty b_n \chi_{[-r_n, -r_n + a_n]}$, where $r_n = 3^{-n}$, $a_n = \frac12 4^{-n}$, $b_n = 2^n$ and $x=0$.
Then for $r=r_n$
$$
\rlap{-}\!\!\int_{B_r(x)} \min \left\{|g(y)-g(x)|,1 \right\} dy = \frac{1}{2 r_n}\sum_{k=n}^\infty a_k = \frac13 \left(\frac34\right)^{n} \to 0
$$
while
$$
\rlap{-}\!\!\int_{B_r(x)} |g(y)-g(x)| \, dy = \frac{1}{2 r_n}\sum_{k=n}^\infty a_k b_k = \frac12 \left(\frac32\right)^{n} \to \infty
$$
as $n\to \infty$.
For the original question one can consider $f(x) = x g(x)$. Let me know if additional details are needed.
Update. For some closely related results see also Proposition 3.65 in Functions of Bounded Variation by L. Ambrosio, N. Fusco and D. Pallara (2000).
