These are both uniform. But the outcome is different
As you originally pointed out, a physical dart can't hit a single point on a number line. It will hit an infinite number of them simultaneously. This is true whether you're worrying about rationals or reals.
But if you have a dart so sharp that its tip is zero-dimensional, one that can hit a single point on a real line, and you throw it at a composite of the rationals from [0,9] and the reals from [9,10], it will have a 10% chance of hitting an irrational number (within [9,10]), and it will have a 90% chance of missing the line entirely, striking one of the holes in the rational interval [0,9]. The chance of hitting a rational number will not improve from 0.
Do you have a model of uniform selection in mind, or do you find that it's easier to say the words without assigning them any particular meaning?
This is all in your head. Who are you responding to? Where did your three claims ("sampling uniformly by distance from 0" / "sampling uniformly by element count" / "randomly sampling the line") come from? What does "sampling uniformly by distance" mean? Uniform sampling is done by count for discrete sets and by area for continua. You have yet to mention a discrete set.