$|\int_D f(x) dx| \leq \int_D |f(x)| dx$ is just the infinitestimal version of the triangle inequality commonly presented in any book on vector spaces
Can we replace the definition of triangle inequality with this version of the statement ($|\int_D f(x) dx| \leq \int_D |f(x)| dx$)?
"Replace" is a strong word. The triangle inequality in normed spaces $(X,\|\cdot\|)$: $$\|x+y\| \leq \|x\|+\|y\|, \quad \forall\,x,y \in X$$ is more general, in the sense that it can take a lot of forms depending on which space and norm you're dealing with, for example: $$\left(\sum_{n \geq 1} |x_n+y_n|^p\right)^{1/p} \leq \left(\sum_{n \geq 1} |x_n|^p\right)^{1/p} + \left(\sum_{n \geq 1} |y_n|^p\right)^{1/p}$$in $\ell^p(\mathbb{N})$ or $$\sup_{n \geq 1}|x_n+y_n| \leq \sup_{n \geq 1}|x_n|+\sup_{n \geq 1}|y_n|$$in $\ell^\infty(\mathbb{N})$ or $$\sum_{k=0}^n \sup_{x \in [a,b]}|f^{(k)}(x)+g^{(k)}(x)| \leq \sum_{k=0}^n \sup_{x \in [a,b]}|f^{(k)}(x)|+\sum_{k=0}^n \sup_{x \in [a,b]}|g^{(k)}(x)|$$in ${\cal C}^n[a,b]$, etc.
The other inequality $\left|\int_Df(x)\,{\rm d}x\right|\leq \int_D|f(x)|\,{\rm d}x$ is just that. However, there is no harm in looking at it as a triangle inequality: you can think of the triangle inequality applied for Riemann sums (for example), and then passing to the limit. I used to confuse $\leq$ with $\geq$ there all the time and had to keep drawing graphs to get it right - once I realized what you did now, it all came more naturally.