It's indeed pretty annoying how many conventions there are. I'll try to be as explicit as possible, but feel free to ask for more details.

In the first expression:

In the first expression:

• $R$ has dimensions of length
• $r$ and $k$ have no dimensions
• $k$ is scaled so it's $-1$, $0$, or $1$

To go from the first expression to the second expression, we define $$r' = r R_0, \quad k' = k / R_0^2, \quad a(t) = R(t) / R_0$$ where I'm denoting the new variables with a prime just to be extra clear. Then

• $r'$ has dimensions of length and $k'$ has dimensions of inverse length squared
• $a$ has no dimensions

These are the two main conventions for the FRW metric. Your third example is a bit unusual, because it tries to set $k'$ to $-1$, $0$, or $1$, which doesn't make any sense because $k'$ has dimensions.

However, $k'$ is a constant, so we can use it to define our unit of length. Taking positive curvature for concreteness, this means that rather than measuring lengths in meters, we measure in a unit system where the numeric value of $k'$ is $1 \, (\text{length unit})^{-2}$. Another way of saying this is that this example is setting $k' = 1$ in the same sense that you may set $c = 1$ in special relativity. You can always do this by defining your unit system appropriately, but then naive dimensional analysis will stop working, and you'll have to put the $k'$'s and $c$'s back in at the end to get valid numbers out.