The natural inclusion of $\mathbb{R}$ in $\mathbb{C}$ is the mapping $$f: \mathbb{R} \to \mathbb{C}, \; x \mapsto x + 0i.$$ Given this, is it completely accurate to say that $x \in \mathbb{C}$? Or would we rather say that we can identity $x$ with an element $x + 0i$ that lives in $\mathbb{C}$? I assume that te former is true, since we do write that $\mathbb{R} \subset \mathbb{C}$, but since the complex numbers are by definition those numbers we can write in the form $a + bi$, I am not completely sure of why this is.
This mapping, in other words, seems less of sending $x$ to $x + 0i$, but rather asserting that $x + 0i = x$, so the identity element in $\mathbb{C}$ is not $0 + 0i$, but rather $0$.
Am I thinking of this correctly?
As you've constructed it, no, it would not be accurate to say $x \in \mathbb{C}$ since you explicitly stated that $f: \mathbb{R} \to \mathbb{C}$.
In response to your closing comments: mappings don't assert anything. They're just instructions for generating something based on what input was provided. (Although the person doing the math could make such an assertion and use that mapping as a persuasive argument.)