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.