All definitions I've seen for the statement "$E,F$ are linearly disjoint extensions of $k$" are only meaningful when $E,F$ are given as subfields of a larger field, say $K$. (I am happy with the equivalence of the various definitions I've seen in this case.)

For example, see Wikipedia says it means the map $E\otimes_k F\to E.F$ is injective, where $E.F$ denotes their compositum (smallest field containing them both) in $K$.

However, I often see the term used when this is not the case, even when the field extensions are not algebraic (so there is no tacit assumption that they live in the algebraic closure).

Examples I can think of right now:

1. Eisenbud, Commutative Algebra, Theorem A.13 (p.564 in my edition) says, in characteristic $p$,

"$K$ is separable over $k$ iff $k^{1/p^{\infty}}$ is linearly disjoint from $K$."

2. Liu, Algebraic Geometry and Arithmetic Curves, Corollary 2.3 (c) (p. 91) says, for an integral algebraic variety $X$ over a field $k$ with function field $K(X)$,

"$X$ is geometrically integral iff $K(X)$ and $\overline{k}$ are linearly disjoint over $k$.

Question: What is the definition of "linearly disjoint" for field extensions which are not specified inside a larger field?

I'm hesitant here to start making things up ad-hoc if there might be some standard interpretation I should know about...