A weaker definition of local non-satiation can also imply indifference "curve"
(2) does not imply (1). Consider a utility function with "circular indifference curves", e.g. $u(x,y)=-(x-1)^2-(y-1)^2$. At the bliss point $(1,1)$, the function satisfies (2) but violates (1).

(2) does imply (3), and the proof should be similar to the one showing that (1) implies (3).