I was astonished to read at Wikipedia that
The term complex representation has slightly different meanings in mathematics and physics. In mathematics, a complex representation is a group representation of a group (or Lie algebra) on a complex vector space. In physics, a complex representation is a group representation of a group (or Lie algebra) on a complex vector space that is neither real nor pseudoreal. In other words, the group elements are expressed as complex matrices, and the complex conjugate of a complex representation is a different, non-equivalent representation.
Are these two definitions somehow related or even equivalent in some sense?
So far, coming from a physics, background my definition of a complex rep was that for a given representation
$$ \pi : \mathfrak{g} \rightarrow gl(V) $$
$\pi(x)$ for $x \in \mathfrak{g}$ is different than $\bar \pi(x)$.
Different means that the conjugate matrices can't be transformed into the non-conjugate matrices by a similarity transformation. I never thought about if in this context $\mathfrak{g}$ must be a real or complex Lie algebra or if $V$ must be a real or a complex vector space in order for the representation to be real or complex.
By definition the complex representation $\bar{\pi}$ acts on $\bar V$ instead of on $V$.
$$ \bar{\pi} : \mathfrak{g} \rightarrow gl(\bar V) $$
Thus if $V$ is a real vector space, we have $\bar V = V$ and this implies $\pi(x) = \bar{\pi}(x)$. Therefore, I think, both definitions are equivalent.
Can someone confirm this or is there some subtle difference?
There is a difference, which is not subtle. It's right there in the definition: the physics definition has the additional condition that the complex conjugate $\overline{V}$ of the representation $V$ isn't isomorphic to it. This happens whenever any value of the character
$$\chi_V(g) = \text{tr}(\pi(g))$$
is not real, since the character of $\overline{V}$ is the complex conjugate of the character of $V$. So, for example, the defining $3$-dimensional complex representation of $SU(3)$ has this property.
Take a look also at the Frobenius-Schur indicator.