For the history, one probably has to distinguish the $C^2$ and $C^1$ case: The $C^2$ case is completely analogous to the finite-dimensional case. But the $C^1$ case does not follow by approximation as in finite dimensions but requires a rather different approach. My guess is that it was indeed one of the 4 mentioned authors who did the $C^1$ case first.

Concerning the “counterexamples” in (i), I would argue that you use the wrong definition of "relatively compact". In a not necessarily Hausdorff space the natural definition is IMHO the following: $M\subseteq X$ is relatively compact in $X$ if there is a compact $K\subseteq X$ with $M\subseteq K$. With this definition, the “counterexamples” in (i) become empty.