abstract algebra - Linearly disjoint vs. free field extensions

Consider two field extensions $K$ and $L$ of a common subfield $k$ and suppose $K$ and $L$ are both subfields of a field $\Omega$, algebraically closed.

Lang defines $K$ and $L$ to be 'linearly disjoint over $k$' if any finite set of elements of $K$ that are linearly independent over $k$ stays linearly independent over $L$ (it is, in fact, a symmetric condition). Similarly, he defines $K$ and $L$ to be 'free over $k$' if any finite set of elements of $K$ that are algebraically independent over $k$ stays algebraically independent over $L$.

He shows right after that if $K$ and $L$ are linearly disjoint over $k$, then they are free over $k$.

Anyway, Wikipedia gives a different definition for linearly disjointness, namely $K$ and $L$ are linearly disjoint over $k$ iff $K \otimes_k L$ is a field, so I was wondering:

do we have a similar description of 'free over $k$' in terms of the tensor product $K \otimes_k L$?

It should be a weaker condition than $K \otimes_k L$ being a field, perhaps it needs to be a integral domain?