Definition of Ohm in SI basic units in words

I think the short answer is, you don't. The reason we call the unit of force a Newton and not a kg m/s$^2$ is because it is convenient and it expresses the relation you want to convey when used elsewhere (e.g., $F=-kx$ for a spring).

Similarly, it is convenient to "hide" the MKS base units into a single term, the potential $V$ in this case, so that the formula is easier to remember and that the relation is conveyed, in this case the relation between potential difference, current, and resistance.

I'm not sure there's much of a point to what you're asking. The intuitive way to understand an Ohm is to use $\Omega = V/A$. If you want to use SI units, you can, and the math indeed tells you that your other definition is correct, but you're not gonna get much out of it. Indeed, the most you could do is to separate it like this:

$$\begin{align}\Omega &= \frac{\text{kg} \cdot \text{m}^2 }{ \text{s}^3 \cdot \text{A}^2 }\\ &= \frac{ \text{kg}\cdot\text{m}^2}{\text{s}^2}\cdot \frac1{\text{A}\cdot\text{s}}\cdot \frac1{\text{A}} \\ &= \frac{\text{J}/\text{C}}{\text{A}}\\ &= \frac{\text{V}}{\text{A}} \end{align} $$

This is just a proof of the equivalence between the two definitions, but don't expect to get any nice word description of the SI definition.

Not sure whether this is correct, but if you have to do it, I think you can say that it is:

the work done by the conductor per unit charge per unit current through the conductor, or in terms of SI units, $\mathrm{\frac JC\cdot \frac1A}$

which is the same as:

the work done by the conductor per unit current per unit time per unit current, $\mathrm{\frac J{A\cdot s\cdot A}}$

We know that the work done is equal to the dot product of the force and the displacement, so it is:

the electric force multiplied by the displacement of the charge carrier per unit time per unit charge squared, $\mathrm{\frac{N\cdot m}{s\cdot A^2}}$

and we know force has SI units $\mathrm{kg\;m\;s^{-2}}$

So I guess you can say that the ohm is the resistance when one newton of electric force causes a charge carrier to displace one meter in one second with a current of one ampere.

I would go on and say that it is the resistance when a charge carrier of one kilogram accelerates at one meter per second squared, and this acceleration causes the charge carrier to displace one meter in one second, producing a current of one ampere. But I'm not very certain about the "charge carrier of one kilogram" part.