I am having problems visualizing the "mechanics" of the Lebesgue integral, but after much editing of the question I think I get it (at least for nice functions where measure theory can be somewhat taken for granted).
So I decided to posted the material I have been working on as a proposed answer.
Part of the misunderstanding had to do with plots found online showing slabs of horizontal, brick-like constructs, as opposed simple functions. In addition, the initial definitions on the chapter on Lebesgue integrals in A Garden of Integrals by Frank E. Burk:
If a function $f$ is bounded measurable on the interval $[a,b]$ with $\alpha<f<\beta$, we can partition the range of $f$: $\alpha=y_0<y_1<\cdots<y_n=\beta$, and denote $E_{\,k}=\{ x \in [a,b] \,|\, y_{\,k-1}\leq f<y_{\,k} \}$, for $k=1, 2, \cdots,n$.
Now we form the lower sum, $\displaystyle\sum_{k=1}^n y_{\,k-1}\, \mu\,(E_{\,k})$, and the upper sum, $\displaystyle\sum_{k=1}^n y_{\,k}\, \mu\,(E_{\,k})$.
Comparing the supremum of the lower sums with the infimum of the upper sum over all possible partitions of $[\alpha,\beta]$, we see if these two numbers are equal, say $A$, we say $f$ is Lebesgue integrable on $[a,b]$, and we write $A=L\displaystyle\int_a^bf\,d\mu$.
... defining the meaning of Lebesgue integrable and not the definition of the Lebesgue integral, led me to take a wrong turn, confusing the Lebesgue integral with the Darboux integral (as pointed out under comments), more akin to a Riemann integral, although Riemann integration does not use lower and upper sums. Here is a graphic representation:
You're confusing the things of which the supremum and infimum are taken.
In your drawing, each box is just a term in the sums.
What you should be taken the sup and inf of is this: consider all the partitions of the interval. Calculate all the upper sums with those partitions and put them in a set. Then calculate the infimum of that set.
Do the same with the lower sums.
If those numbers are equal, then the integral exists.