[Note: I interpreted the question as how to explain the relationship between the two definitions]
Let's first consider the underlying algebra before turning to real-world models.
As an example, suppose we have four reals $a_1 < a_2 < a_3 <a_4$ with mean $a$ between $a_2$ and $a_3.\,$ Here the equivalence connecting the two views of the mean is as follows
$$\overbrace{a+a+a+a = a_1 + a_2 + a_3 + a_4}^{\large\text{definition of mean } a}\iff \overbrace{a\!-a_1\, +\, a\!-a_2\, =\, a_3 - a\, +\, a_4 -a}^{\large{ a_i\ \text{are balanced around the mean } a}} $$
The direction $(\Rightarrow)$ arises as follows: in the definition, we can cancel a LHS $a$ from each $a_i \ge a$ on the RHS, and we can cancel each RHS $a_i < a$ from an $a$ on the LHS, yielding the equivalent balanced form on the right. This method can be used more generally to check an equality of sums by rewriting it more simply, e.g.
$\qquad\qquad\qquad\begin{align}
&\color{#c00}{222}+\color{#0a0}{200}+1000+4119\\
=\ &\color{#c00}{100}+\color{#0a0}{328}+2113+3000\end{align}$
$\ \iff\ \begin{align}
&\color{#c00}{122} + 1119\\
=\ &\color{#0a0}{128} + 1113
\end{align}$
$\iff \begin{align}
&6\\
=\ &6
\end{align}$
Above we cancelled $\color{#c00}{100}$ from both sides, leaving the summand $\color{#c00}{122}$ on the new LHS.
Next, $\, $ we cancelled $\color{#0a0}{200}$ from both sides, leaving the summand $\color{#0a0}{128}$ on the new RHS, etc. The OP is just the special case when all summands are equal on one side, say $a$. This replaces each $a_i$ by its distance from $a$, doing it on the equation side that keeps all the summands nonnegative.
Conversely, direction $(\Leftarrow)$ follows by inverting the prior, i.e. by adding terms to both sides of the equation in order to move all negated terms to the opposite side of the equation, i.e. by eliminating all subtractions.
A simple real-world model is an old-fashioned balance scale. Then the prior additions and subtractions to both sides amount to adding or removing equal weights from both sides of the scale - which preserves balance. One can illustrate the above equivalence by explicitly performing the steps in the above sketched proof using weight manipulations. Of course one should use much smaller numbers than I did above (I chose those larger numbers only to highlight that the method can yield nonntrivial simplifications).
