Although this is a question about what's a continuous random variable, it seems that there are at least 2 definitions being used.
I'm interested in understanding the consequences (limitations/advantages) for using each one, and maybe someone knows other definitions and add them together with an explanation of their consequences.
On
If you consider the measure induced by a random variable $X$, i.e. $\mu(A) := P\{X \in A\}$ for $A$ in the Borel sigma-algebra, then you would say $X$ is a continuous random variable if $\mu$ is absolutely continuous with respect to the Lebesgue measure.
That is if the Lebesgue measure of some measurable set $A$ is $0$, then it must also be true that $\mu(A) = 0$. If $A$ is a singleton, then you know that its Lebesgue measure is $0$. Hence, $X$ cannot have positive probability at any single point if it is continuous. So this is in line with our intuition of how a continuous random variable should look.
If $\mu$ is absolutely continuous with respect to the Lebesgue measure, then by the Radon-Nikodym theorem there exists a function $f$ such that
$$\mu(A) = \int_A f\,dx $$
You also know that considering sets of the form $A := (-\infty,z]$ is sufficient to define $\mu$ uniquely. Hence, if we can find a function $f$ such that the following holds for every $z \in \mathbb{R}$ we can declare $X$ to be continuous. $$F(z) := \mu((-\infty,z]) = \int_{-\infty}^z f\,dx $$ It is no coincidence that $F$ in your second definition is called an absolutely continuous function. You can read more on this at https://en.wikipedia.org/wiki/Absolute_continuity
The first definition is the one that is standard. In some pre-measure theory courses the second definition is given and it is hard to see the difference between the two without aknowledge of measure theory. However, the difference between the two is important and anyone who wants to learn modern probability theory should adapt definition 1).