How to adjust for an additional variable.

In the second example, for the case of Virginia, you used 58 fatal car crashes, when that was the number for Vermont. That was likely a typo.

Now, it can not be the second option you give. Notice the number of crashes ends up cancelling so we end up ignoring that information altogether.

I believe you are interested in answering how many car crashes occur (in a region) per car density. In other words, you want to compute an incidence proportion, adjusted for car density. In this regard, I think you have the right answer. That said, there is a reason why incidence rates are adjusted 'per 1000 people' (or 10.000 people) and I don't think that applies for your problem.

The reason why incidence proportions are adjusted is that it is very difficult to communicate (and make sense of) very small numbers, as incidence proportions typically are. Instead, we adjust it by computing it in '$1000$ people' units. For example, it would be difficult to communicate to an audience that the proportion incidence of COVID is $0.001$. It is even difficult to read it aloud and it would be difficult to compare it to, say, an incidence of $0.00023$, for example.

Instead, we read COVID incidence as 100 cases per $100.000$ persons, and that is something most people can understand, even though it is the same rate and it is mathematically trivial to go from one to the other.

To further clarify my position, I suggest a dimension analysis:

Define Vermont's car density as
$$\text{Vermont car density}:= \frac{250,000 \text{ cars}}{642,495 \text{ persons}} = 0.3891081 \frac{\text{cars}}{\text{person}}$$ And now define Vermont's rate of fatal car crashes per car density:
$$\text{Vermont rate of fatal car crashes per car density}:=$$ $$\frac{\text{Vermont's fatal car crashes}}{\text{Vermont's car density}} = \frac{ 58 \text{ fatal fatal car crashes}}{0.3891081 \frac{\text{cars}}{\text{person}}} = 14905.9 \frac{\text{ fatal car crashes}}{\frac{\text{cars}}{\text{person}}}$$ One could read this result as 'Vermont has a rate of car fatal crashes per car density of $14905$ fatal car crashes-person per car.
Because the units for this rate of car crashes per car density are already large enough, I do not think it is warranted to transform them into 'per 1000'. If we insisted on measuring this in 'per 1000 people' it would read as $14905900$ fatal car crashes-person per car, which becomes harder to read and to communicate.

Edit:


Vermont: 58/(250,000/642,495)

Virginia: 796/(2,500,000 / 8,632,044)