What does Rosenlicht mean by a "point"? By $k(v_1,v_2)$?

Question

This is cross-posted from Math.SE at the recommendation of a commenter.

I'm reading M. Rosenlicht's 1956 paper, "Some Basic Theorems on Algebraic Groups" [link], and having trouble with some of the language and notation. In particular, I have two questions:

1) Rosenlicht defines an algebraic group as a variety with a group structure on the points. He pointedly does not assume the ground field $k$ is algebraically closed. As he was writing prior to Grothendieck's development of scheme theory, I am not sure how to interpret "points". In this context, does he mean the $k$-points? The $\overline k$-points i.e. the geometric points? Or what?

2) With $G$ an algebraic group, and $V$ a variety with an action of $G$, all defined over a field $k$, Rosenlicht frequently writes expressions like "$k(g,v)$" with $g\in G, v\in V$; for example, on p. 403, we have

We say $G$ operates on $V$ (or that $V$ is a pre-transformation space for $G$) if for each component $G_\alpha$ of $G$ we are given a rational map $g\times v\to g(v)$ of $G_\alpha\times V\rightarrow V$ such that if $k$ is a field of definition for $G$, $V$, and each of these rational maps and if $g_1\times g_2 \times v$ is a generic point over $k$ of $G_\alpha \times G_\beta \times V$ ($G_\alpha$, $G_\beta$ being any components of $V$) then

$$(1)\:\:\:\: g_1(g_2(v)) = g_1g_2(v).$$ $$(2)\:\:\:\: k(g_1,g_1(v)) = k(g_1,v).$$

What's meant by $k(g_1,v)$ here? I want it to be a residue field but it has 2 arguments. Is it the composite of the two residue fields? (In which case, the answer to question 1 cannot be "$k$-points"?)

Aside: I am particularly interested in Theorem 2 on p. 407 ("Rosenlicht's theorem"). However, my goal is to be able to read Rosenlicht's paper, so I would prefer a gloss of his usages over a reference to a more modern statement and proof.

Addendum: The accepted answer contains a link to another question/answer which links to this very helpful piece by Raynaud in the Sept. 1999 Notices, discussing Weil's foundational approach (which turns out to be the context for Rosenlicht's paper). I include the link here for the sake of self-containedness of the question/answer pair.

Answer 1

Weil's foundations (which Rosenlicht uses) use a universal domain, which is an algebraically closed extension of $k$ of infinite transcendence degree. There is a discussion of universal domains here. In this language, "point" means a point with coordinates in a universal domain. The expression $k(g,v)$ means the subfield of the universal domain generated over $k$ by the coordinates of the points $g$ and $v$.